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+{"text":"\\section{Introduction} \\label{sec:intro}\n\nLet $X$ be an algebraic curve with an action of a finite group $G$ over an algebraically closed field $k$.\nStudying the $k[G]$-module structure of cohomologies of $X$ is a natural and fundamental topic in algebraic geometry.\nIn  the  classical  case,  that  is, over the field of complex numbers, the equivariant structure\nof the module of holomorphic differentials was completely determined  by  Chevalley and Weil, cf. \\cite{Chevalley_Weil_Uber_verhalten}. Their result remains valid when $\\cha k \\nmid \\# G$. \\\\\n\nWhen $\\cha k = p > 0$ and $p| \\# G$, the structure of $H^0(X, \\Omega_{X\/k})$ becomes much more complicated. Most of previous results on this subject concern either the tame ramification case (cf. e.g. \\cite{Kani_Galois_module}, \\cite{Nakajima_Galois_module}, \\cite{Kock_galois_structure}, \\cite{chinburg_epsilon_constants}) or focus on some special\ngroups (see e.g. \\cite{Valentini_Madan_Automorphisms} for the case of cyclic groups, \\cite{WardMarques_HoloDiffs} for abelian groups or~\\cite{Bleher_Chinburg_Kontogeorgis_Galois_structure} for groups with a cyclic Sylow subgroup).\nOther results compute the structure of the space of global sections of\nan invertible sheaf of sufficiently large degree (cf. \\cite{Borne_Cohomology_of_G_sheaves}).\nEven less is known about the equivariant structure of the de Rham cohomology.\nIn the tame ramification case the de Rham--Euler characteristic class has been computed in the $K$-theory category (cf.~\\cite{Chinburg1994} and~\\cite{chinburg_epsilon_constants}).\nThere are also results concerning the de Rham cohomology of Deligne--Lusztig curves (cf. \\cite{Lusztig_Coxeter_orbits}), of Hermitian curves (cf. \\cite{Dummigan_95}, \\cite{Dummigan_99} and \\cite{Haastert_Jantzen_Filtrations})\nand of Suzuki curves (cf. \\cite{Liu_decomposition_suz}, \\cite[p. 2535]{Gross_Rigid_local_systems_Gm} and \\cite{Malmskog_Pries_Weir_dR_Suzuki}). In general, it is known that the Hodge--de Rham exact sequence does not split (see \\cite{Hortsch_canonical_representation}, \\cite{KockTait2018} for explicit examples and the next section for my result that yields a more precise criterion).\nOne should also mention, that there are several results\nconcerning the structure of $H^1_{dR}(X\/k)$ as a Dieudonn\\'{e} module for a curve~$X$ with a $\\ZZ\/p$-action, cf. \\cite{PriesZhu_p_rank_AS} and~\\cite{Booher_Cais_a_numbers}.\n\nIn this paper we are interested in the case when $\\cha k = p > 0$ and $G$ is a finite $p$-group.\nLet $\\pi : X \\to Y := X\/G$ be the canonical morphism.\nFor any ${P \\in X(k)}$ denote by $G_{P, i}$ the $i$-th ramification group\nat $P$ and let:\n\\[\nd_{P}:= \\sum_{i \\ge 0} (\\# G_{P, i} - 1), \\quad \nd_{P}' := \\sum_{i \\ge 1} (\\# G_{P, i} - 1), \\quad\nd_{P}'' := \\sum_{i \\ge 2} (\\# G_{P, i} - 1)\n\\]\n(note that $d_P$ is the different exponent at $P$). We assume that the cover $\\pi$ satisfies the following assumptions:\n\\begin{enumerate}[(A)]\n\t\\item \\label{enum:A} $G_P$ (the stabilizer of $P$ in $G$) is a normal subgroup of $G$ for every $P \\in X(k)$,\n\t\n\t\\item \\label{enum:B} there exists a function $z \\in k(X)$ (a ``magical element'') satisfying $\\ord_P(z) \\ge -d_P'$\n\tfor every $P \\in X(k)$ and $\\tr_{X\/Y}(z) \\neq 0$.\n\n\\end{enumerate}\nBy the assumption~\\ref{enum:A}, for $Q \\in Y(k)$ we may denote $G_Q := G_P$ and $d_Q := d_P$\nfor any $P \\in \\pi^{-1}(Q)$. Let $I_G := \\{ \\sum_{g \\in G} a_g g \\in k[G] : \\sum_{g \\in G} a_g = 0 \\}$ be the augmentation ideal of the group~$G$\nand let $J_G := I_G^{\\vee}$ be the dual of $I_G$. For any subgroup $H \\le G$\nwe consider also the relative augmentation ideal $I_{G, H} := \\Ind^G_H \\, I_H$\nand its dual $J_{G, H} := \\Ind^G_H \\, J_H$. Finally, the $k[G]$-modules\n$I_{X\/Y}$ and $J_{X\/Y}$ are defined by:\n\\begin{align*}\n\tI_{X\/Y} &:= \\ker \\left( \\sum : \\bigoplus_{Q \\in Y(k)} I_{G, G_Q} \\to I_G \\right),\\\\\n\tJ_{X\/Y} &:= \\coker \\left( \\diag : J_G \\to  \\bigoplus_{Q \\in Y(k)} J_{G, G_Q} \\right).\n\\end{align*}\n\\begin{Theorem} \\label{thm:main_thm}\n\tSuppose that $G$ is a finite $p$-group and $k$ is an algebraically closed field of characteristic $p$.\n\tLet $\\pi : X \\to Y$ be a $G$-cover of smooth projective curves over~$k$, satisfying conditions~\\ref{enum:A} and \\ref{enum:B}. Denote by $g_Y$ the genus of $Y$ and by $B \\subset Y(k)$ -- the branch locus of $\\pi$.\n\tThen we have the following isomorphisms of $k[G]$-modules:\n\n\t\\begin{align*}\n\t\tH^0(X, \\Omega_{X\/k}) &\\cong k[G]^{\\oplus g_Y} \\oplus I_{X\/Y} \\oplus \\bigoplus_{Q \\in B} H^0_Q,\\\\\n\t\tH^1(X, \\mc O_X) &\\cong k[G]^{\\oplus g_Y} \\oplus J_{X\/Y} \\oplus \\bigoplus_{Q \\in B} H^1_Q,\\\\\n\t\tH^1_{dR}(X\/k) &\\cong k[G]^{\\oplus 2 \\cdot g_Y} \\oplus I_{X\/Y} \\oplus J_{X\/Y} \n\t\t\\oplus \\bigoplus_{Q \\in B} H^1_{dR, Q},\n\t\\end{align*}\n\n\twhere $H^0_Q$, $H^1_Q$, $H^1_{dR, Q}$ are certain $k[G]$-modules that depend only on the rings $\\mc O_{X, Q}$ and on the element $z$ (see~\\eqref{eqn:H0Q}, \\eqref{eqn:H1Q} and~\\eqref{eqn:H1dRQ}\n\tfor precise definitions).\n\n\\end{Theorem}\nWe now give some motivation for Theorem~\\ref{thm:main_thm}. Since the Hodge--de Rham spectral sequence of~$X$\ndegenerates at the first page, one obtains the Hodge--de Rham exact sequence:\n\\begin{equation} \\label{eqn:intro_hodge_de_rham_se}\n\t0 \\to H^0(X, \\Omega_{X\/k}) \\to H^1_{dR}(X\/k) \\to H^1(X, \\mc O_X) \\to 0.\n\\end{equation}\nAs shown in \\cite[Proposition 3.1]{Garnek_equivariant}, under some mild assumptions the 'defect':\n\\begin{equation*}\n\t\\dim_k H^0(X, \\Omega_{X\/k})^G + \\dim_k H^1(X, \\mc O_X)^G - \\dim_k H^1_{dR}(X\/k)^G\n\\end{equation*}\nis a sum of local terms indexed by points of $X$. In particular, if the exact sequence~\\eqref{eqn:intro_hodge_de_rham_se} splits equivariantly,\nthe action of $G$ on $X$ is weakly ramified, i.e. $G_{P, 2} = 0$ for every $P \\in X(k)$, cf. \\cite[Main Theorem]{Garnek_equivariant}.\nThis result is somehow surprising, since it shows that a global condition (i.e. splitting of the Hodge--de Rham exact sequence) is affected by a local condition (i.e. vanishing of ramification groups).\\\\\n\nOne can hope that this observation is a part of a bigger picture, namely that\n$H^0(X, \\Omega_{X\/k})$, $H^1(X, \\mc O_X)$ and $H^1_{dR}(X\/k)$ decompose into\ncertain global and local parts. The global part should depend on the ``topology'' of the cover (i.e.\non the branch locus and inertia groups) and not on the higher ramification groups. Moreover, the global part of $H^0(X, \\Omega_{X\/k}) \\oplus H^1(X, \\mc O_X)$ should be isomorphic to the global part of $H^1_{dR}(X\/k)$. \nFinally, one can expect that the local parts depend only on the local rings of the\nfixed points of the action of $G$. Theorem~\\ref{thm:main_thm} proves a slightly weaker statement.\nConjecturally, the local terms  $H^0_Q$, $H^1_Q$, $H^1_{dR, Q}$ do not depend on the element $z$ from the condition~\\ref{enum:B}. We expect also that $\\dim_k H^0_Q = \\frac{1}{2} \\# \\pi^{-1}(Q) \\cdot d_Q''$.\\\\\n\nA result of Elkin and Pries yields a similar decomposition of $H^0(X, \\Omega_{X\/k})$ and $H^1_{dR}(X\/k)$ into global and local parts in the case when $Y = \\PP^1$ and $p = 2$ (cf.~\\cite[Theorem 1.2]{elkin_pries_ekedahl_oort}). However, the context is different -- they study the $k[F, V]$-module structure of the mentioned groups (where $F$ and $V$ denote the Frobenius and Verschiebung morphisms).\nExplicit examples show that such a decomposition is impossible in the category of $k[F, V]$-modules in general, cf. \\cite[Example~7.2]{Booher_Cais_a_numbers}.\\\\\n\nAs an application of Theorem~\\ref{thm:main_thm}, we give a description of cohomologies of $\\ZZ\/p$-covers (cf. Section~\\ref{sec:Zp}). \nLet $J_i$ denote the unique indecomposable $k[\\ZZ\/p]$-module of dimension~$i$ for $i = 1, \\ldots, p$.\n\\begin{Corollary} \\label{cor:cohomology_of_Zp}\n\tSuppose that $k$ is an algebraically closed field of characteristic~$p$.\n\tLet $\\pi : X \\to Y$ be a $\\ZZ\/p$-cover of smooth projective curves over $k$.\n\tSuppose that $\\pi$ has a global standard form (cf. Subsection~\\ref{subsec:gsf}). Then, as $k[\\ZZ\/p]$-modules:\n\n\t\\begin{align*}\n\t\tH^0(X, \\Omega_{X\/k}) &\\cong H^1(X, \\mc O_X) \\cong J_{p}^{\\oplus g_Y} \\oplus J_{p-1}^{\\oplus (\\# B - 1)} \\oplus\n\t\t\\bigoplus_{i = 1}^{p-1} J_i^{\\oplus \\alpha(i)}, \\\\\n\t\tH^1_{dR}(X\/k) &\\cong J_{p}^{\\oplus 2 \\cdot g_Y} \\oplus J_{p-1}^{\\oplus \\alpha},\n\t\\end{align*}\n\n\twhere $g_Y$ is the genus of $Y$, $B \\subset Y(k)$ denotes the branch locus of $\\pi$, $m_Q := d_Q'\/(p-1)$ and:\n\n\t\\begin{align*}\n\t\n\t\t\\alpha(i) &:= \\sum_{Q \\in B} \\left( \\left \\lceil \\frac{m_Q \\cdot (i+1)}{p} \\right \\rceil\n\t\t- \\left \\lceil \\frac{m_Q \\cdot i}{p} \\right \\rceil \\right),\\\\\n\t\t\\alpha &:= 2 \\cdot (\\# B - 1) + \\sum_{Q \\in B} (m_Q - 1).\n\t\\end{align*}\n\n\\end{Corollary}\nFor $H^0(X, \\Omega_{X\/k})$ this result was already known (cf.~\\cite[Theorem~1]{Valentini_Madan_Automorphisms}), however we don't know of any previous results regarding the de Rham cohomology.\\\\\n\nTheorem~\\ref{thm:main_thm} allows us to give a converse of~\\cite[Main Theorem]{Garnek_equivariant}\nfor covers satisfying~\\ref{enum:A} and~\\ref{enum:B}.\n\\begin{Corollary} \\label{cor:hdr_exact_sequence}\n\n\tKeep assumptions of Theorem~\\ref{thm:main_thm}. Then the action of $G$ on $X$ is weakly ramified\n\tif and only if there is an isomorphism of $k[G]$-modules:\n\n\t\\[ H^1_{dR}(X\/k) \\cong H^0(X, \\Omega_{X\/k}) \\oplus H^1(X, \\mc O_X). \\]\n\n\n\\end{Corollary}\nIt is reasonable to ask how often are the conditions~\\ref{enum:A} and~\\ref{enum:B} satisfied.\nUnfortunately, not every $p$-group cover has a magical element, cf. Subsection~\\ref{subsec:no_magical_element} for a counterexample.\nHowever, it turns out that a generic $G$-cover satisfies~\\ref{enum:A} and~\\ref{enum:B}. To be precise,\nlet $k$ be an algebraically closed field of characteristic $p$. Fix a finite $p$-group~$G$ and an affine open subset $U$ of a smooth projective curve $Y$ over $k$. Let $M_{U, G}$ denote the moduli space\nof pointed $G$-covers of $Y$ unramified over $U$, as defined in~\\cite{Harbater_moduli_of_p_covers}.\n\n\\begin{Theorem} \\label{thm:generic_intro}\n\tThe set of covers satisfying~\\ref{enum:A} and~\\ref{enum:B} forms a dense subset of~$M_{U, G}$.\n\\end{Theorem}\n\\noindent The proof of Theorem~\\ref{thm:generic_intro} gives a way to construct inductively\ncovers satisfying~\\ref{enum:A} and~\\ref{enum:B}.\n\\begin{Example} \\label{ex:intro}\n\n\tLet $Y$ be the elliptic curve:\n\n\t\\begin{equation*}\n\t\tY : w^2 = (x - \\alpha_1) \\cdot (x - \\alpha_2) \\cdot (x - \\alpha_3)\n\t\\end{equation*}\n\n\tover $k$.  Define $X$ to be a smooth projective curve with the function field given by $k(X) = k(Y)(y_1, y_2, y_3)$, where:\n\n\t\\begin{align*}\n\t\ty_1^p - y_1 &= (x - \\alpha_1)^{-a_1},\\\\\n\t\ty_2^p - y_2 &= (x - \\alpha_1)^{-a_2} \\cdot (x - \\alpha_2)^{-b_2},\\\\\n\t\ty_3^p - y_3 &= (x - \\alpha_1)^{-a_3} \\cdot (x - \\alpha_2)^{-b_3} \\cdot (x - \\alpha_3)^{-c_3}\\\\\n\t\t&+ (y_1 + y_2) \\cdot (x - \\alpha_1)^{-a_2} \\cdot (x - \\alpha_2)^{-b_2},\n\t\\end{align*}\n\n\tthe natural numbers $a_1, a_2, b_2, a_3, b_3, c_3$ are non-divisible by $p$ and satisfy:\n\n\t\\[\n\ta_1 > p, \\quad a_2 > 4a_1 \\cdot p, \\quad a_3 > 4 \\cdot p \\cdot (a_1 + a_2 + b_2), \\quad b_3 > 4 b_2.\n\t\\]\n\n\tThen $X \\to Y$ is an $E(p^3)$-cover satisfying~\\ref{enum:A} and~\\ref{enum:B},\n\twhere $E(p^3)$ denotes the Heisenberg group $\\textrm{mod } p$ (see Corollary~\\ref{cor:example_from_intro} for the details).\n\n\\end{Example}\nOur results leave some questions open. In which generality does Theorem~\\ref{thm:main_thm} hold? How to describe\nthe modules $H^0_Q$, $H^1_Q$, $H^1_{dR, Q}$ without using the magical element? \nIs it possible to generalize the above considerations to higher dimensional varieties or to crystalline\ncohomology? We plan to investigate those questions in the near future.\n\\subsection*{Strategy of the proof of Theorem~\\ref{thm:main_thm}}\nWe explain now the idea behind the proof of Theorem~\\ref{thm:main_thm} for the module of holomorphic differentials. The proof for $H^1(X, \\mc O_X)$ and\n$H^1_{dR}(X\/k)$ follows the same strategy. Our approach is divided into two main steps.\nIn both steps we use the fact that the function $z$ is a normal\nelement of $k(X)\/k(Y)$.\\\\\n\nIn the first step we compare the module $H^0(X, \\Omega_{X\/k})$ with $H^0(X, \\Omega_{X\/k}(R))$, the module of differentials with logarithmic poles in the ramification locus of the cover (cf. Proposition~\\ref{prop:log_diffs_and_diffs}). To this end we write any form $\\omega \\in \\Omega_{k(X)\/k}$ as $\\sum_{g \\in G} g^*(z) \\omega_g$ for $\\omega_g \\in \\Omega_{k(Y)\/k}$ and\nconsider the following question.\n\\begin{Question}\n\tSuppose that $\\omega \\in H^0(X, \\Omega_{X\/k})$. What are the possible values of the residues $\\res_Q(\\omega_g)$ for $Q \\in B$ and $g \\in G$? In other words, what is the image of $H^0(X, \\Omega_{X\/k})$ under the map:\n\n\t\\[\n\t\\res_G : \\Omega_{k(X)\/k} \\to \\bigoplus_{Q \\in B} k[G], \\qquad \\omega \\mapsto \\sum_{Q \\in B} \\sum_{g \\in G} \\res_Q(\\omega_g) g_Q,\n\t\\]\n\n\twhere $g_Q \\in \\bigoplus_{B} k[G]$ is the element with $g$ on the $Q$-th component and $0$ on other components?\n\\end{Question}\n\\noindent There are two conditions imposed on the residues of $\\omega_g$:\n\\begin{enumerate}[(1)]\n\t\\item The first one follows from the residue theorem: $\\sum_{Q \\in B} \\res_Q(\\omega_g) = 0$.\n\t\n\t\\item The second one is imposed by the condition $\\res_P(\\omega) = 0$ for $P \\in \\pi^{-1}(Q)$.\n\\end{enumerate}\nThe conditions (1) and (2) define the module $I_{X\/Y}$. In this way we obtain an equivariant homomorphism\n\\begin{equation} \\label{eqn:map_to_IXY}\n\t\\res_G : H^0(X, \\Omega_{X\/k}) \\to I_{X\/Y}.\n\\end{equation}\nIt turns out that this homomorphism is split. One constructs\nits section by choosing appropriate forms in $\\Omega_{k(Y)\/k}$ with known residues.\nWe can apply a similar reasoning for logarithmic differential forms, neglecting the condition (2).\nIn this way one obtains a split homomorphism:\n\\begin{equation} \\label{eqn:map_to_kGB}\n\t\\res_G : H^0(X, \\Omega_{X\/k}(R)) \\to k[G]_B,\n\\end{equation}\nwhere $k[G]_B$ is the submodule of $\\bigoplus_B k[G]$ defined by the condition (1). It turns out that the maps~\\eqref{eqn:map_to_IXY} and~\\eqref{eqn:map_to_kGB}\nhave the same kernel (cf. Proposition~\\ref{prop:log_diffs_and_diffs}).\\\\\n\nIn the second step we observe that we have an inclusion of sheaves of the same rank:\n\\[\n\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B) \\subset \\Omega_{X\/k}(R).\n\\]\nHence, their quotient $\\mc T$\nis a torsion sheaf and its global sections decompose as a sum of local parts. By\napplying the long exact sequence to the sequence of sheaves:\n\\[\n0 \\to \\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B) \\to \\Omega_{X\/k}(R) \\to \\mc T \\to 0\n\\]\nand observing that $k[G]$ is an injective $k[G]$-module we find the $k[G]$-structure on $H^0(X, \\Omega_{X\/k}(R))$.\n\\subsection*{Constructing magical elements}\nWe discuss now main ideas behind the proof of Theorem~\\ref{thm:generic_intro}.\nSuppose that $X \\to Y$ factors through a Galois cover $X' \\to Y$.\nIt turns out that if both $X \\to X'$ and $X' \\to Y$ have magical elements, then\n$X \\to Y$ also has a magical element (cf. Lemma~\\ref{lem:new_magical_elts}). Hence everything comes down to\nconstructing magical elements for $\\ZZ\/p$-covers.\nWe prove that a global standard form of a $\\ZZ\/p$-cover (cf. Subsection~\\ref{subsec:gsf} for a definition)\nyields a magical element. Moreover, every sufficiently ramified $\\ZZ\/p$-cover\nhas a global standard form (cf. Lemma~\\ref{lem:criterion_for_gsf}).\nTheorem~\\ref{thm:generic_intro} follows by noting that a generic $G$-cover can be factored\ninto sufficiently ramified $\\ZZ\/p$-covers.\n\\subsection*{Outline of the paper}\nIn Section~\\ref{sec:notation} we give necessary notation. Section~\\ref{sec:magical_elements}\nproves some properties of the magical element $z$ from the condition~\\ref{enum:B}.\nThis allows us to prove the part of Theorem~\\ref{thm:main_thm} concerning the module of holomorphic differentials\nand the cohomology of the structure sheaf in Section~\\ref{sec:OmegaX}. We prove\nthe decomposition of the de Rham cohomology from the Theorem~\\ref{thm:main_thm} in Section~\\ref{sec:dR}.\nIn Section~\\ref{sec:AS_covers} we introduce the notion of a global standard form\nof a $\\ZZ\/p$-cover. This allows us to construct a magical element for a large class\nof Artin--Schreier covers. Also, we prove Corollary~\\ref{cor:cohomology_of_Zp}.\nFinally, in Section~\\ref{sec:constructing_magical_elements} we prove that a \ngeneric $p$-group cover has a magical element and discuss Example~\\ref{ex:intro}.\n\n\\subsection*{Acknowledgements}\nThe author wishes to express his gratitude to Bartosz Naskr\u0119cki and Wojciech Gajda,\nwhose comments helped to considerably improve the exposition of the paper.\nThe ``global--local'' point of view on this problem was inspired by a conversation with\nPiotr Achinger in November 2018.\n\n\\section{Notation} \\label{sec:notation}\nIn this subsection we introduce some notation concerning algebraic curves.\nFor an arbitrary smooth projective curve $Y$ over a field $k$ we denote by $k(Y)$ the function field of $Y$.\nAlso, we write $\\ord_Q(f)$ for the order of vanishing of a function $f \\in k(Y)$ at a point $Q \\in Y(k)$.\nLet $\\mf m_{Y, Q}^n := \\{ f \\in k(Y) : \\ord_Q(f) \\ge n \\}$ for any $n \\in \\ZZ$.\nWe will often identify a finite set $S \\subset Y(k)$ with a reduced divisor in $\\Divv(Y)$.\nThus e.g. $\\Omega_{Y\/k}(S)$ will denote the sheaf of logarithmic differential\nforms with poles in $S$.\\\\\n\nLet $G$ be a finite group and $\\pi : X \\to Y$ be a finite separable $G$-cover of smooth projective curves over a field $k$.\nIn the sequel we identify $\\Omega_{k(Y)\/k}$ with a submodule of $\\Omega_{k(X)\/k}$ and\n$k(Y)$ with a subfield of $k(X)$. We denote the ramification index of $\\pi$ at $P \\in X(k)$ by $e_{X\/Y, P}$ and\nby $G_{P, i}$ -- the $i$-th ramification group of $\\pi$ at $P$, i.e.\n\\[\nG_{P, i} := \\{ \\sigma \\in G : \\sigma(f) \\equiv f \\pmod{\\mf m_P^{i+1}} \\quad \\forall_{f \\in \\mc O_{X, P}} \\}.\n\\]\nAlso, we use the following notation:\n\\begin{align*}\n\td_{X\/Y, P}:= \\sum_{i \\ge 0} (\\# G_{P, i} - 1), \\quad\n\td_{X\/Y, P}' := \\sum_{i \\ge 1} (\\# G_{P, i} - 1), \\quad\n\td_{X\/Y, P}'' := \\sum_{i \\ge 2} (\\# G_{P, i} - 1)\n\\end{align*}\n($d_{X\/Y, P}$ is the different exponent at $P$).\nRecall that for any $P \\in X(k)$ and $\\omega \\in \\Omega_{k(Y)\/k}$:\n\\begin{equation} \\label{eqn:valuation_of_diff_form}\n\t\\ord_P(\\omega) = e_{X\/Y, P} \\cdot \\ord_{\\pi(P)}(\\omega) + d_{X\/Y, P}.\n\\end{equation}\nFor any sheaf $\\mc F$ on $X$ and $Q \\in Y(k)$ we abbreviate $(\\pi_* \\mc F)_Q$ to $\\mc F_Q$.\nWe write briefly $\\tr_{X\/Y}$ for the trace\n\\[\n\t\\tr_{k(X)\/k(Y)} : k(X) \\to k(Y).\n\\]\nNote that it induces a map $\\Omega_{k(X)\/k} \\to \\Omega_{k(Y)\/k}$, which we also denote by $\\tr_{X\/Y}$.\nFor a future use we note the following properties of trace:\n\\begin{itemize}\n\t\\item For $f \\in k(X)$ and $Q \\in Y(k)$:\n\n\t\\begin{equation} \\label{eqn:valuation_of_trace}\n\t\t\\tr_{X\/Y}(f) \\in \\mf m_{Y, Q}^{\\alpha},\n\t\\end{equation}\n\n\twhere $\\alpha := \\min \\{ [(\\ord_P(f)+d_{X\/Y, P})\/e_{X\/Y, P}] : P \\in \\pi^{-1}(Q) \\}$.\n\t\n\t\\item[] For the proof recall that by \\cite[Lemma 5.4 (4)]{Mollin_ANT} for any ideal\n\t$J$ of $\\mc O_{Y, Q}$:\n\n\t\\begin{align*}\n\t\t\\tr_{X\/Y}(f \\mc O_{X, Q}) \\subset J &\\Leftrightarrow f \\mc O_{X, Q} \\subset J \\mc D_{X\/Y}^{-1}\\\\\n\t\t&\\Leftrightarrow e_{X\/Y, P} \\cdot \\ord_Q(J) - d_{X\/Y, P} \\le \\ord_P(f) \\quad \\forall_{P \\in \\pi^{-1}(Q)}\\\\\n\t\t&\\Leftrightarrow \\ord_P(J) \\le \\left[\\frac{\\ord_P(f) + d_{X\/Y, P}}{e_{X\/Y, P}} \\right]  \\quad \\forall_{P \\in \\pi^{-1}(Q)}\\\\\n\t\t&\\Leftrightarrow \\mf m_{Y, Q}^{\\alpha} \\subset J,\n\t\\end{align*}\n\n\twhere $\\mc D_{X\/Y}$ is the different ideal of the extension $\\mc O_{X, Q}\/\\mc O_{Y, Q}$. \n\n\t\\item Let $S := \\pi^{-1}(Q)$. Then:\n\n\t\\begin{equation} \\label{eqn:trace_and_diff_forms}\n\t\t\\tr_{X\/Y}(\\Omega_{X\/k, Q}) \\subset \\Omega_{Y\/k, Q} \\quad \\textrm{ and }\n\t\t\\tr_{X\/Y}(\\Omega_{X\/k}(S)_{Q}) \\subset \\Omega_{Y\/k}(\\{ Q \\})_Q.\n\t\\end{equation}\n\n\t\\item[] Indeed, the first part of~\\eqref{eqn:trace_and_diff_forms} follows by the main result of~\\cite{Zannier_traces_diff_forms}.\n\tThe second part is immediate by using the first part and noting that for any $f \\in k(X)$, $f \\neq 0$:\n\n\t\\[\n\t\\tr_{X\/Y}(df\/f) = \\frac{d(N_{X\/Y} f)}{N_{X\/Y} f},\n\t\\]\n\n\twhere $N_{X\/Y} : k(X) \\to k(Y)$ is the norm of the extension of fields $k(X)\/k(Y)$.\n\n\t\\item For any $\\eta \\in \\Omega_{k(X)\/k}$ and $Q \\in Y(k)$:\n\n\t\\begin{equation} \\label{eqn:residue_and_trace}\n\t\t\\sum_{P \\in \\pi^{-1}(Q)} \\res_P(\\eta) = \\res_Q(\\tr_{X\/Y}(\\eta))\n\t\\end{equation}\n\n\t\\item[] (see \\cite[Proposition 1.6]{Hubl_residual_representation} or \\cite[p.~154, $(R_6)$]{Tate_residues_differentials_curves}).\n\\end{itemize}\nIn the most part of the article we will assume that $k$, $X$, $Y$ and $G$ are as in Theorem~\\ref{thm:main_thm}.\nIn this situation we adapt the following notation:\n\\begin{itemize}\n\t\\item $B \\subset Y(k)$ -- the set of branch points of $\\pi$,\n\t\n\t\\item $R \\subset X(k)$ -- the set of ramification points of $\\pi$,\n\t\n\t\\item $U := Y \\setminus B$, $V := \\pi^{-1}(U)$,\n\t\n\t\\item $e_P := e_{X\/Y, P}$, $d_P := d_{X\/Y, P}$, $d_P' := d_{X\/Y, P}'$, $d_P'' := d_{X\/Y, P}''$ for $P \\in X(k)$,\n\n\t\\item $X_P := X\/G_{P, 0}$ is the quotient curve. We denote the image of $P$ on $X_P$ by $\\ol P$.\n\tSimilarly, for set $S \\subset X(k)$, we write $\\ol S$ for its image on $X_P$.\n\\end{itemize}\nAlso, by abuse of notation, for $Q \\in Y(k)$ we write $G_{Q, i} := G_{P, i}$, $e_Q := e_P$, $d_Q := d_P$, $X_Q := X_P$ etc. for any $P \\in \\pi^{-1}(Q)$. Note that these quantities don't depend on the choice\nof $P$. \\\\\n\nRecall that the map:\n\\[\nk(X) \\times G \\to k(X), \\qquad f \\cdot g := g^*(f)\n\\]\ninduces a natural right action on $k(X)$, since $g_1^* \\circ g_2^* = (g_2 \\cdot g_1)^*$ for any $g_1, g_2 \\in G$.\nSimilarly, we have the structure of a right $k[G]$-module on $H^0(X, \\Omega_{X\/k})$, $H^1(X, \\mc O_X)$, $H^1_{dR}(X\/k)$, etc. Note also that $G_P = G_{P, 0} = G_{P, 1}$, since $k$ is algebraically closed of characteristic $p$ and $G$ is a $p$-group (cf.~\\cite[Corollary 4.2.3., p. 67]{Serre1979}).\n\n\\section{Magical elements} \\label{sec:magical_elements}\nKeep the assumptions of Theorem~\\ref{thm:main_thm}. In this section we study the properties of the magical element $z \\in k(X)$\nsatisfying the condition~\\ref{enum:B}.\\\\\nIt turns out that the condition $\\tr_{X\/Y}(z) \\neq 0$ guarantees that $z$ is a normal element,\nsee e.g.~\\cite[Theorem 1]{Childs_Orzech_On_modular}. We give a proof for completeness.\n\\begin{Proposition} \\label{prop:g(z)_is_a_basis}\n\tThe set $\\{ g^*(z) : g \\in G \\}$ is a $k(Y)$-basis of $k(X)$.\n\\end{Proposition}\n\\begin{proof}\n\n\tThe proof is based on the following identity in the ring $\\FF_p[x_g : g \\in G]$:\n\n\t\\begin{equation} \\label{eqn:G-determinant}\n\t\t\\det [x_{gh}]_{g, h \\in G} = \\left(\\sum_{g \\in G} x_g \\right)^{\\# G}.\n\t\\end{equation}\n\n\tThis formula is just a version of the Group Determinant Formula\n\t(cf.~\\cite[p. 71]{Washington_Intro_to_cyclotomic}) in the case of positive characteristic (see~\\cite[Lemma 2.2 and Corollary 2.4]{Huynh_Artin_Schreier_extensions}).\n\tWe show that the set $\\{ g^*(z) : g \\in G \\}$ is linearly independent over $k(Y)$.\n\tSuppose that for some $f_g \\in k(Y)$:\n\n\t\\begin{equation*}\n\t\t0 = \\sum_{g \\in G} g^*(z) \\cdot f_g.\n\t\\end{equation*}\n\n\tThen for any $h \\in G$:\n\n\t\\begin{equation*}\n\t\t0 = \\sum_{g \\in G} h^*(g^*(z)) \\cdot f_g = \\sum_{g \\in G} (gh)^*(z) \\cdot f_g.\n\t\\end{equation*}\n\n\tHowever, by~\\eqref{eqn:G-determinant}:\n\n\t\\[\n\t\\det[(gh)^*(z)] = \\tr_{X\/Y}(z)^{\\# G} \\neq 0\n\t\\]\n\n\tand hence $f_g = 0$ for all $g \\in G$. This ends the proof.\n\\end{proof}\nNote that $\\Omega_{k(X)\/k}$ is a rank one $k(X)$-module and similarly $\\Omega_{k(Y)\/k}$ is a rank one $k(Y)$-module.\nHence, in the light of Proposition~\\ref{prop:g(z)_is_a_basis}, for any $\\omega \\in \\Omega_{k(X)\/k}$ there exists\na unique system of differential forms $(\\omega_g)_{g \\in G}$ in $\\Omega_{k(Y)\/k}$ for which\n\\[\n\\omega = \\sum_{g \\in G} g^*(z) \\omega_g.\n\\]\nNote that for any $g, h \\in G$:\n\\begin{equation} \\label{eqn:g_component_of_h}\n\th(\\omega)_g = \\omega_{g h^{-1}}.\n\\end{equation}\nIndeed:\n\\begin{align*}\n\th^*(\\omega) = \\sum_{g \\in G} h^*(g^*(z)) \\omega_g = \\sum_{g \\in G} (g \\cdot h)^*(z) \\omega_g\n\t= \\sum_{g \\in G} g^*(z) \\omega_{gh^{-1}}.\n\\end{align*}\n\\begin{Lemma} \\label{lem:inclusions_of_modules}\n\tWe have the following inclusions of sheaves on $Y$:\n\n\t\\begin{align*}\n\t\t\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k} \\subset \\pi_* \\Omega_{X\/k},\\\\\n\t\t\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B) \\subset \\pi_* \\Omega_{X\/k}(R).\n\t\\end{align*}\n\n\\end{Lemma}\n\\begin{proof}\n\tNote that by~\\eqref{eqn:valuation_of_diff_form} for any $Q \\in Y(k)$, $\\omega \\in \\Omega_{Y\/k, Q}$, $g \\in G$ and $P \\in \\pi^{-1}(Q)$:\n\n\t\\begin{equation*}\n\t\t\\ord_P(g^*(z) \\omega) \\ge -d_P' + e_P \\cdot \\ord_Q(\\omega) + d_P \\ge -d_P' + d_P = e_P - 1 \\ge 0.\n\t\\end{equation*}\n\n\tThe first inclusion follows. The second inclusion may be proven analogously.\n\\end{proof}\nObserve that $\\pi_* \\Omega_{X\/k}(R)$ and $\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B)$ are coherent sheaves\nof rank $\\# G$. Therefore, their quotient is torsion and thus is isomorphic to:\n\\begin{equation} \\label{eqn:quotient_differentials}\n\t\\pi_* \\Omega_{X\/k}(R)\/\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B) \\cong\n\t\\bigoplus_{Q \\in Y(k)} i_{Q, *}(H^0_Q) \n\\end{equation}\nwhere $i_Q : \\Spec \\mc O_{Y, Q} \\to Y$ and:\n\\begin{equation} \\label{eqn:H0Q}\n\tH^0_Q := \\Omega_{X\/k}(R)_Q\/\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B)_Q.\n\\end{equation}\n\\begin{Lemma} \\label{lem:zQ_regular}\n\n\t$z_Q := \\tr_{X\/X_Q}(z)  \\in \\mc O_{X_Q, Q}$.\n\n\\end{Lemma}\n\\begin{proof}\n\n\tFix a point $P \\in \\pi^{-1}(Q)$. Observe that $X_Q \\to Y$ is unramified over~$Q$. Hence by~\\eqref{eqn:valuation_of_trace}:\n\n\t\\[\n\t\t\\tr_{X\/X_Q}(z) \\in \\mf m_{X_Q, \\ol P}^{[(-d_P' + d_P)\/e_P]} = \\mc O_{X_Q, \\ol P}.\n\t\\]\n\n\tThis finishes the proof.\n\\end{proof}\nRecall that the dual basis of $\\{ g^*(z) : g \\in G \\}$ with respect to the trace map is of\nthe form $\\{ g^*(z^{\\vee}) : g \\in G \\}$ for some $z^{\\vee} \\in k(X)$ (cf.~\\cite[Theorem~3.13.19]{Hachenberger_Jungnickel_Topics}). By definition, $z^{\\vee}$ satisfies\nfor any $g_1, g_2 \\in G$:\n\\begin{equation} \\label{eqn:def_of_dual_elt}\n\t\\tr_{X\/Y}(g_1(z) \\cdot g_2(z^{\\vee})) = \n\t\\begin{cases}\n\t\t1, & g_1 = g_2,\\\\\n\t\t0, & g_1 \\neq g_2.\n\t\\end{cases}\n\\end{equation}\nFor a future use note also that\n\\begin{equation} \\label{eqn:trace_of_dual_z}\n\t\\tr_{X\/Y}(z), \\, \\tr_{X\/Y}(z^{\\vee}) \\in k^{\\times}.\n\\end{equation}\nIndeed, by~\\eqref{eqn:valuation_of_trace} for every $Q \\in Y(k)$ one has $\\tr_{X\/Y}(z) \\in \\mc O_{Y, Q}$.\nThis yields $\\tr_{X\/Y}(z) \\in \\bigcap_{Q \\in Y(k)} \\mc O_{Y, Q} = H^0(Y, \\mc O_Y) = k$ and \n$\\tr_{X\/Y}(z) \\in k^{\\times}$. Moreover by~\\eqref{eqn:def_of_dual_elt}:\n\\begin{align*}\n\t\\tr_{X\/Y}(z^{\\vee}) \n\t&= \\frac{\\tr_{X\/Y} \\left(z^{\\vee} \\cdot \\tr_{X\/Y}(z) \\right)}{\\tr_{X\/Y}(z)}\n\t= \\frac{\\tr_{X\/Y} \\left(z^{\\vee} \\cdot \\sum_{g \\in G} g^*(z) \\right)}{\\tr_{X\/Y}(z)}\\\\\n\t&= \\frac{\\sum_{g \\in G} \\tr_{X\/Y}(z^{\\vee} \\cdot g^*(z))}{\\tr_{X\/Y}(z)} = \\frac{1}{\\tr_{X\/Y}(z)} \\in k^{\\times}.\n\\end{align*}\t\nFor any $f \\in k(X)$, we will denote by $(f_g)_{g \\in G}$ the unique system of functions $f_g \\in k(Y)$ such that:\n\\[\nf = \\sum_{g \\in G} g^*(z^{\\vee}) f_g.\n\\]\nNote that by~\\eqref{eqn:def_of_dual_elt} for any $g \\in G$, $\\omega \\in \\Omega_{k(X)\/k}$ and $f \\in k(X)$:\n\\begin{align}\n\t\\tr_{X\/Y}(g^*(z^{\\vee}) \\cdot \\omega) = \\omega_g \\quad \\textrm{ and } \\quad\n\t\\tr_{X\/Y}(g^*(z) \\cdot f) = f_g. \\label{eqn:gth_component_trace_f}\n\\end{align}\n\n\\begin{Lemma} \\label{lem:inclusions_of_modules2}\n\tWe have the following inclusions:\n\n\t\\begin{align*}\n\t\t\\pi_* \\mc O_X &\\subset \\bigoplus_{g \\in G} g^*(z^{\\vee}) \\mc O_Y,\\\\\n\t\t\\pi_* \\mc O_X(-R) &\\subset \\bigoplus_{g \\in G} g^*(z^{\\vee}) \\mc O_Y(-B).\n\t\\end{align*}\n\n\\end{Lemma}\n\\begin{proof}\n\n\tSuppose that $Q \\in Y(k)$ and $f \\in \\mc O_{X, Q}$. Then, for any $\\omega \\in \\Omega_{Y\/k, Q}$, using~\\eqref{eqn:gth_component_trace_f} and~\\eqref{eqn:residue_and_trace}:\n\n\t\\begin{align*}\n\t\t\\res_Q(f_g \\cdot \\omega) &= \\res_Q(\\tr_{X\/Y}(g^*(z) f) \\cdot \\omega)\\\\\n\t\t&= \\sum_{P \\in \\pi^{-1}(Q)} \\res_P(f  \\cdot  g^*(z) \\cdot \\omega)\n\t\t= 0,\n\t\\end{align*}\n\n\twhere the last equality follows, since $f \\in \\mc O_{X, Q}$ and $g^*(z) \\cdot \\omega \\in \\Omega_{X\/k, Q}$ by Lemma~\\ref{lem:inclusions_of_modules}. Hence $f_g \\in \\mc O_{X, Q}$.\n\tThe second inclusion follows analogously.\n\\end{proof}\nLemma~\\ref{lem:inclusions_of_modules2} implies that:\n\\begin{equation*} \\label{eqn:quotient_functions}\n\t\\frac{\\bigoplus_{g \\in G} g^*(z^{\\vee}) \\mc O_Y(-B)}{\\pi_* \\mc O_X(-R)} \\cong \\bigoplus_{Q \\in Y(k)} i_{Q, *}(H^1_Q),\n\\end{equation*}\nwhere:\n\\begin{equation} \\label{eqn:H1Q}\n\tH^1_Q := \\bigoplus_{g \\in G} g^*(z^{\\vee}) \\mc O_Y(-B)_Q\/\\mc O_X(-R)_Q. \n\\end{equation}\n\\begin{Lemma} \\label{lem:properties_H0Q_H1Q}\n\tLet $H^0_Q$, $H^1_Q$ be defined by~\\eqref{eqn:H0Q} and~\\eqref{eqn:H1Q}.\n\t\\begin{enumerate}[(1)]\n\t\t\\item $H^1_Q$ is dual to $H^0_Q$ as a $k[G]$-module.\n\t\t\n\t\t\\item If $d_Q'' = 0$ then $H^0_Q = H^1_Q = 0$.\n\t\t\n\t\t\\item $\\sum_{Q \\in B} \\dim_k H^0_Q = \\sum_{Q \\in B} \\dim_k H^1_Q = \\sum_{Q \\in B} \\frac{1}{2} d_Q'' \\cdot \\#\\pi^{-1}(Q)$.\n\t\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n\t(1) One checks that the duality pairing is induced by:\n\t\n\t\t\\begin{align*}\n\t\t\t\\Omega_{X\/k}(R)_Q \\times \\bigoplus_{g \\in G} g^*(z^{\\vee}) \\mc O_Y(-B)_Q &\\to k,\\\\\n\t\t\t(\\omega, f) &\\mapsto \\sum_{Q \\in B} \\sum_{g \\in G} \\res_Q(\\omega_g \\cdot f_g).\n\t\t\\end{align*}\n\t\n\t\tWe omit the details.\n\t\t\n\t(2) Suppose that $d_Q'' = 0$. Then $d_Q = 2 \\cdot (e_Q - 1)$ and $d_Q' = (e_Q - 1)$.\n\t\tWrite:\n\t\n\t\t\\begin{equation*}\n\t\t\tz^{\\vee} = \\sum_{g \\in G} g^*(z) \\cdot f_g \\qquad \\textrm{ for } f_g \\in k(Y).\n\t\t\\end{equation*}\n\t\n\t\tThen~\\eqref{eqn:def_of_dual_elt} yields $f_g = \\det[a_{h_1 h_2}]_{h_1, h_2 \\in G}\/\\det[\\tr_{X\/Y}(h_1(z) \\cdot h_2(z))]_{h_1, h_2 \\in G}$, where:\n\t\t\\[\n\t\ta_{h_1 h_2} :=\n\t\t\\begin{cases}\n\t\t\ttr_{X\/Y}(h_1(z) \\cdot h_2(z)), & h_1 \\neq g,\\\\\n\t\t\t\\delta_{h_2 g}, & h_1 = g.\t\n\t\t\\end{cases}\n\t\t\\]\n\t\n\t\tBut~the Group Determinant Formula~\\eqref{eqn:G-determinant} easily implies that\n\t\n\t\t\\begin{align*}\n\t\t\t\\det[\\tr_{X\/Y}(h_1(z) \\cdot h_2(z))]_{h_1, h_2 \\in G} &= \n\t\t\t\\det[\\tr_{X\/Y}((h_1 \\cdot h_2')(z) \\cdot z)]_{h_1, h_2' \\in G}\\\\\n\t\t\t&= \\tr_{X\/Y}(z)^{2 \\cdot \\# G} \\in k^{\\times}.\n\t\t\\end{align*}\n\t\n\t\tMoreover, \\eqref{eqn:valuation_of_trace} yields that \n\t\n\t\t\\[\n\t\ttr_{X\/Y}(h_1(z) \\cdot h_2(z)) \\in \\mf m_{Y, Q}^{\\left[\\frac{-2d'_Q + d_Q}{e_Q} \\right]} = \\mc O_{X, Q}.\n\t\t\\]\n\t\n\t\tHence $f_g \\in \\mc O_{X, Q}$ and $\\ord_Q(z^{\\vee}) \\ge \\ord_Q(z) = - d_Q'$.\n\t\tThus\n\t\tif $f \\in \\mc O_Y(-B)_Q$, then for any $P \\in \\pi^{-1}(Q)$:\n\t\n\t\t\\[\n\t\t\\ord_P(g^*(z^{\\vee}) \\cdot f) \\ge -d_Q' + e_Q = 1\n\t\t\\]\n\t\n\t\tand $g^*(z^{\\vee}) \\cdot f \\in \\mc O_X(-R)_Q$.\n\t\tIt follows that $H^1_Q = 0$ and thus also $H^0_Q = 0$ by~(1).\n\t\t\n\t(3) This follows from~\\eqref{eqn:quotient_differentials} by taking global sections and applying the Riemann--Hurwitz formula and Riemann--Roch theorem.\n\\end{proof}\nWe end this section by giving some necessary conditions for $\\pi$ to have a magical element.\nOne of the conditions will play a role in the proof of Theorem~\\ref{thm:main_thm} in Section~\\ref{sec:dR}.\n\\begin{Lemma} \\label{lem:GME_implies_no_etale_cover}\n\tKeep assumptions of Theorem~\\ref{thm:main_thm}. Then:\n\t\\begin{enumerate}[(1)]\n\t\t\\item $\\pi$ does not factor through an \\'{e}tale morphism $X' \\to Y$\n\t\tof degree $> 1$,\n\t\t\n\t\t\\item $\\langle G_Q : Q \\in B \\rangle = G$. \n\t\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n\t\\begin{enumerate}[(1)]\n\t\t\\item Suppose to the contrary that $\\pi$ factors through a non-trivial \\'{e}tale morphism $X' \\to Y$.\n\t\tThen $z$ is also a magical element for the cover $X \\to X'$. Hence $\\tr_{X\/X'}(z) \\in k^{\\times}$.\n\t\tBut then:\n\t\n\t\t\\[\n\t\t\\tr_{X\/Y}(z) = \\tr_{X'\/Y}(\\tr_{X\/X'}(z)) = [k(X') : k(Y)] \\cdot \\tr_{X\/X'}(z) = 0.\n\t\t\\]\n\t\n\t\tContradiction ends the proof.\n\t\t\n\t\t\\item Note that $X' := X\/\\langle G_Q : Q \\in B \\rangle \\to Y$ is an \\'{e}tale subcover\n\t\tof $\\pi$. Thus by~(1) it must be of degree $1$ and $G = \\langle G_Q : Q \\in B \\rangle$.\n\t\\end{enumerate}\n\\end{proof}\n\n\n\\section{Holomorphic differentials} \\label{sec:OmegaX}\nThe goal of this section is to prove the part of Theorem~\\ref{thm:main_thm} concerning $H^0(X, \\Omega_{X\/k})$ and $H^1(X, \\mc O_X)$. The first step in this direction is to compare holomorphic and logarithmic differentials. This is achieved by the following Proposition.\n\\begin{Proposition} \\label{prop:log_diffs_and_diffs}\n\n\tKeep assumptions of Theorem~\\ref{thm:main_thm}. \n\tWe have the following isomorphism of right $k[G]$-modules:\n\t\\[\n\tH^0(X, \\Omega_{X\/k}) \\oplus k[G]^{\\# B - 1} \\cong H^0(X, \\Omega_{X\/k}(R)) \\oplus I_{X\/Y}.\n\t\\]\n\n\\end{Proposition}\nWe first show how Proposition~\\ref{prop:log_diffs_and_diffs} implies the part of Theorem~\\ref{thm:main_thm}\nconcerning $H^0(X, \\Omega_{X\/k})$ and $H^1(X, \\mc O_X)$.\n\\begin{proof}[Proof of Theorem~\\ref{thm:main_thm}, part 1]\n\tRecall that by~\\eqref{eqn:quotient_differentials} we have an exact sequence of $\\mc O_Y$-modules:\n\n\t\\[\n\t0 \\to \\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B) \\to \\pi_* \\Omega_{X\/k}(R) \\to \\bigoplus_{Q \\in Y(k)} i_{Q, *}(H^0_Q) \\to 0.\n\t\\]\n\n\tMoreover, by Serre's duality and Riemann--Roch theorem (cf.~\\cite[Corollary III.7.7, Theorem IV.1.3]{Hartshorne1977}):\n\t\\begin{align*}\n\t\th^0(Y, \\Omega_{Y\/k}(B)) &= g_Y + \\# B - 1,\\\\\n\t\th^1(Y, \\Omega_{Y\/k}(B)) &= 0.\n\t\\end{align*}\n\n\tHence, after taking sections:\n\n\t\\[\n\t0 \\to k[G]^{g_Y + \\# B - 1} \\to H^0(X, \\Omega_{X\/k}(R)) \\to \\bigoplus_{Q \\in B} H^0_Q \\to 0.\n\t\\]\n\n\tNote that $k[G]$ is injective as a $k[G]$-module (see~\\cite[Corollary 8.5.3]{Webb_finite_group_representations}). Hence:\n\n\t\\[\n\tH^0(X, \\Omega_{X\/k}(R)) \\cong k[G]^{g_Y + \\# B - 1} \\oplus \\bigoplus_{Q \\in B} H^0_Q.\n\t\\]\n\n\tWe combine this with Proposition~\\ref{prop:log_diffs_and_diffs} to obtain:\n\n\t\\begin{equation} \\label{eqn:proof_of_MT1_before_dividing}\n\t\tH^0(X, \\Omega_{X\/k}) \\oplus k[G]^{\\# B - 1} \\cong k[G]^{g_Y + \\# B - 1} \\oplus I_{X\/Y} \\oplus \\bigoplus_{Q \\in B} H^0_Q.\n\t\\end{equation}\n\n\tSince every $k[G]$-module has a unique decomposition into indecomposable $k[G]$-modules (cf. \\cite[Corollary~11.1.7.]{Webb_finite_group_representations}), we may divide both sides\n\tof~\\eqref{eqn:proof_of_MT1_before_dividing} by $k[G]^{\\# B - 1}$. In this way we obtain the\n\tpart of Theorem~\\ref{thm:main_thm} concerning $H^0(X, \\Omega_{X\/k})$. Finally, by Serre's duality,\n\t$H^1(X, \\mc O_X)$ is the dual of $H^0(X, \\Omega_{X\/k})$. This immediately implies the part\n\tof Theorem~\\ref{thm:main_thm} concerning the cohomology of the structure sheaf.\n\\end{proof}\nThe proof of Proposition~\\ref{prop:log_diffs_and_diffs} will occupy the rest of this section.\nIn the sequel we will need the relative augmentation ideal $I_{G, H}$ (as defined in Section~1),\nwhere $H$ is a normal subgroup of $G$. We identify it with:\n\\[\nI_{G, H} = \\left\\{ \\sum_{g \\in G} a_g g \\in k[G] : \\sum_{g \\in g_0 H} a_g = 0 \\quad \\forall_{g_0 \\in G} \\right\\}.\n\\]\nDefine also $k[G]_B := \\ker \\left( \\sum : \\bigoplus_B k[G] \\to k[G] \\right)$ (note that $k[G]_B \\cong k[G]^{\\# B - 1}$ as a $k[G]$-module). For any $Q \\in B$ and $g \\in G$, let $g_Q \\in \\bigoplus_{B} k[G]$ be the element with $g$ on the $Q$-th component and $0$ on other components.\\\\\n\nRecall that in order to prove Proposition~\\ref{prop:log_diffs_and_diffs} we study the image\nof $H^0(X, \\Omega_{X\/k})$ under the map $\\res_G$. Let $\\omega \\in H^0(X, \\Omega_{X\/k})$. Suppose for simplicity that $\\pi^{-1}(Q) = \\{ P \\}$ and\n$\\tr_{X\/Y}(z) = 1$. Then by~\\eqref{eqn:residue_and_trace} $\\res_P(g^*(z) \\omega_g) = \\res_Q(\\omega_g)$. Hence:\n\\begin{align*}\n\t0 = \\res_P(\\omega) = \\sum_{g \\in G} \\res_Q(\\omega_g)\n\\end{align*}\nand $\\sum_{g \\in G} \\res_Q(\\omega_g) \\cdot g \\in I_G$. In general, if $G_Q \\neq G$, $\\sum_{g \\in G} \\res_Q(\\omega_g) \\cdot g \\in I_{G, G_Q}$, which is a consequence of the following lemma.\n\\begin{Lemma} \\label{lem:main_lemma_Omega_Y}\n\tKeep assumptions of Theorem~\\ref{thm:main_thm} and let $Q \\in B$.\n\n\t\\begin{enumerate}[(1), labelindent=0pt, itemindent=0pt, labelwidth=!]\n\t\t\\item For every $\\omega \\in \\Omega_{k(X)\/k}$ and $g_0 \\in G$, the form\n\t\n\t\t\\[\n\t\t\t\\sum_{g \\in g_0 G_Q} \\omega_g\n\t\t\\]\n\t\n\t\tcan be expressed as a combination of forms $g^*(\\tr_{X\/X_Q}(\\omega))$ for $g \\in G$ with coefficients\n\t\tin $\\mc O_{X_Q, Q}$.\n\n\t\t\\item For every $\\omega \\in \\Omega_{X\/k}(R)_Q$ one has:\n\t\n\t\t\\[\n\t\t\t\\omega \\in \\Omega_{X\/k, Q} \\quad \\Leftrightarrow \\quad \\forall_{g_0 \\in G} \\, \\sum_{g \\in g_0 G_Q} \\res_Q(\\omega_g) = 0.\n\t\t\\]\n\t\n\t\\end{enumerate}\n\n\\end{Lemma}\n\\begin{proof}\n\n\tFix a point $P \\in \\pi^{-1}(Q)$. Let $G\/G_Q = \\{ g_1 G_Q, \\ldots, g_r G_Q \\}$ and $P_i := g_i(P)$.\n\tDefine $\\omega_i := \\sum_{g \\in g_i G_Q} \\omega_g$ for $i = 1, \\ldots, r$.\n\t\n\t(1) Observe that:\n\t\n\t\t\\[\n\t\t\\tr_{X\/X_Q}(\\omega) = \\sum_{i = 1}^r g_i^*(z_Q) \\cdot \\omega_i.\n\t\t\\]\n\t\n\t\tThis implies that:\n\t\n\t\t\\begin{equation} \\label{eqn:system_eqns_tr_eta}\n\t\t\tg_j^*(\\tr_{X\/X_Q}(\\omega)) = \\sum_{i = 1}^r (g_i \\cdot g_j)^*(z_Q) \\cdot \\omega_i\n\t\t\\end{equation}\n\t\n\t\tfor every $j = 1, \\ldots, r$. By the Group Determinant Formula~\\eqref{eqn:G-determinant} for the group $G\/G_Q$:\n\t\n\t\t\\[\n\t\t\\det[(g_i \\cdot g_j)^*(z_Q)] = \\left(\\sum_{i=1}^r g_i^*(z_Q) \\right)^{\\# G\/G_Q} = \\tr_{X\/Y}(z)^{\\# G\/G_Q} \\in k^{\\times}.\n\t\t\\]\n\t\n\t\tTherefore the system of linear equations~\\eqref{eqn:system_eqns_tr_eta} and Lemma~\\ref{lem:zQ_regular} imply that $\\omega_i$ can\n\t\tbe expressed as a combination of forms $g_j^*(\\tr_{X\/X_Q}(\\omega))$ with coefficients in $\\mc O_{X_Q, Q}$.\\\\\n\t\t\n\t(2) If $\\omega \\in \\Omega_{X\/k, Q}$ then $\\res_Q(\\omega_i) = 0$ for $i = 1, \\ldots, r$ by part~(1) and~\\eqref{eqn:trace_and_diff_forms}.\n\tSuppose now that $\\omega \\in \\Omega_{X\/k}(R)_Q$ and $\\res_Q(\\omega_i) = 0$ for $i = 1, \\ldots, r$.\n\t\tNote that $\\tr_{X\/X_Q}(\\omega) \\in \\Omega_{X\/k}(\\ol{R})_{Q}$ by~\\eqref{eqn:trace_and_diff_forms}. Therefore, by (1) we have\n\t\t$\\omega_i \\in \\Omega_{X_Q\/k}(\\ol R)_Q \\cap \\Omega_{k(Y)\/k} = \\Omega_{Y\/k}(B)_Q$ and\n\t\t$\\res_Q(\\omega_i) = 0$. Hence $\\omega_i$ is holomorphic at $Q$ (it is a logarithmic form\n\t\twith vanishing residues). \n\t\tIt follows that:\n\t\n\t\t\\[\n\t\t\\tr_{X\/X_Q}(\\omega) = \\sum_{i = 1}^r g_i^*(z_Q) \\cdot \\omega_i \\in \\Omega_{X_Q\/k, Q}.\n\t\t\\]\n\t\n\t\tHence, using~\\eqref{eqn:residue_and_trace}, $\\res_{P_j}(\\omega) = \\res_{\\ol{P}_j}(\\tr_{X\/X_Q}(\\omega)) = 0$\n\t\tfor every $j$. Therefore $\\omega$ must be holomorphic.\t\n\n\\end{proof}\nThe following exact sequence enables to construct various differential forms on $Y$ from ``local data''\n(cf. \\cite[III.7]{Hartshorne1977}):\n\\begin{align} \\label{eqn:constructing_diff_forms}\n\t0 \\to \\Omega_{Y\/k}(Y) \\to \\Omega_{k(Y)\/k} \\to \\bigoplus_{Q \\in Y(k)} \\frac{\\Omega_{k(Y)\/k}}{\\Omega_{Y, Q}}\n\t&\\to \\, \\, k \\to 0,\\\\\n\t(\\omega_Q)_Q &\\mapsto \\sum_{Q \\in Y(k)} \\res_Q(\\omega_Q). \\nonumber\n\\end{align}\nFix a point $Q_0 \\in B$. Let for any $Q \\in B$, $Q \\neq Q_0$, $\\eta_Q \\in H^0(Y, \\Omega_{Y\/k}(Q_0+Q))$ be a fixed\ndifferential form satisfying:\n\\[\n\\res_Q(\\eta_Q) = 1, \\quad \\res_{Q_0}(\\eta_Q) = -1\n\\]\n(note that such a form exists by~\\eqref{eqn:constructing_diff_forms}). Denote also $\\eta_{Q_0} = 0$.\\\\\n\\subsection*{Proof of {Proposition~\\ref{prop:log_diffs_and_diffs}}}\nThe proof is divided into four steps.\nIn the Steps I--III we define auxiliary maps that will be used to construct the isomorphism in Step IV.\nWe abbreviate $\\sum_{Q \\in B} \\sum_{g \\in G}$ to $\\sum_{Q, g}$.\n\\subsection*{Step I} The map $\\res_G$ defines $k[G]$-linear homomorphisms:\n\\[\nH^0(X, \\Omega_{X\/k}) \\to I_{X\/Y} \\quad \\textrm{ and } \\quad H^0(X, \\Omega_{X\/k}(R)) \\to k[G]_B.\n\\]\n\\begin{proof}[Proof of Step I]\n\tWe check now that $\\res_G$ defines a map $H^0(X, \\Omega_{X\/k}) \\to I_{X\/Y}$.\n\tIndeed, for any $Q \\in B$, $\\sum_{g \\in G} \\res_Q(\\omega_g) g \\in I_{G, G_Q}$ by Lemma~\\ref{lem:main_lemma_Omega_Y}~(2). \n\tMoreover:\n\n\t\\[\n\t\\sum_{Q, g} \\res_Q(\\omega_g) g = \\sum_{g \\in G} g \\sum_{Q \\in B} \\res_Q(\\omega_g) = 0\n\t\\]\n\n\tby the residue theorem. It follows that the image of $H^0(X, \\Omega_{X\/k})$ is contained in $I_{X\/Y}$. The $k$-linearity is easy to check. The map in question\n\tis $G$-equivariant, since for any $h \\in G$, the form $\\omega \\cdot h = h^*(\\omega)$ maps to:\n\n\t\\begin{align*}\n\t\t\\sum_{Q, g} \\res_Q((h^*\\omega)_g) \\cdot g_Q\n\t\t&= \\sum_{Q, g} \\res_Q(\\omega_{gh^{-1}}) \\cdot g_Q\\\\\n\t\t&=\\sum_{Q, g} \\res_Q(\\omega_g) \\cdot (g \\cdot h)_Q\\\\\n\t\t&= \\left( \\sum_{Q, g} \\res_Q(\\omega_{g}) \\cdot g_Q \\right) \\cdot h.\n\t\\end{align*}\n\n\t(here we used~\\eqref{eqn:g_component_of_h}).\n\tOne proves that $\\res_G$ defines a map $H^0(X, \\Omega_{X\/k}(R)) \\to k[G]_B$ by applying residue theorem in a similar manner.\n\\end{proof}\n\\subsection*{Step II}\nThe map:\n\\[\n\\bigoplus_{Q \\in B} k[G] \\to \\Omega_{k(X)\/k}, \\quad\n\\sum_{Q, g} a_{Q, g} g_Q \\mapsto \\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q\n\\]\ninduces $k[G]$-linear homomorphisms:\n\\[\nI_{X\/Y} \\to H^0(X, \\Omega_{X\/k}) \\quad \\textrm{ and } \\quad k[G]_B \\to H^0(X, \\Omega_{X\/k}(R)).\n\\]\n\\begin{proof}[Proof of Step II]\n\tLet $\\sum_{Q, g} a_{Q, g} g_Q \\in I_{X\/Y}$ and $\\omega := \\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q$. For $Q \\neq Q_0$, $\\omega \\in \\Omega_{X\/k, Q}$ by Lemma~\\ref{lem:main_lemma_Omega_Y} (2). Moreover, for any $g_0 \\in G$:\n\n\t\\begin{align*}\n\t\t\\sum_{g \\in g_0 G_{Q_0}} \\res_{Q_0}(\\omega_g)\n\t\t= \\sum_{g \\in g_0 G_{Q_0}} \\sum_{Q \\neq Q_0} -a_{Q, g}\n\t\t= \\sum_{g \\in g_0 G_{Q_0}} a_{Q_0, g} = 0.\n\t\\end{align*}\n\n\tHence, by Lemma~\\ref{lem:main_lemma_Omega_Y}~(2), $\\omega$\n\tis regular also over $Q_0$ and thus is an element of $H^0(X, \\Omega_{X\/k})$.\n\tThe $k[G]$-linearity is easy to check. One checks that the map $k[G]_B \\to H^0(X, \\Omega_{X\/k}(R))$\n\tis well-defined in a similar manner.\n\\end{proof}\n\\subsection*{Step III}\nLet $\\omega \\in H^0(X, \\Omega_{X\/k}(R))$. Then:\n\\[\t\\omega^{\\circ} := \\omega - \\sum_{Q, g} \\res_Q(\\omega_g) g^*(z) \\eta_Q \\in H^0(X, \\Omega_{X\/k}). \\]\nMoreover, the map $\\omega \\mapsto \\omega^{\\circ}$ is $k[G]$-linear.\n\\begin{proof}[Proof of Step III]\n\tBy Lemma~\\ref{lem:main_lemma_Omega_Y}~(2) $\\omega^{\\circ}$ is holomorphic at every $Q \\neq Q_0$, since $\\res_Q(\\omega^{\\circ}_g) = \\res_Q(\\omega_g - \\res_Q(\\omega_g) \\cdot \\eta_Q) = 0$ for every $g \\in G$.\n\tMoreover, by the residue theorem:\n\n\t\\begin{equation*}\n\t\t\\res_{Q_0}(\\omega^{\\circ}_g) = -\\sum_{Q \\neq Q_0} \\res_Q(\\omega^{\\circ}_g) = 0.\n\t\\end{equation*}\n\n\tHence $\\omega^{\\circ}$ holomorphic at $Q_0$ by Lemma~\\ref{lem:main_lemma_Omega_Y}~(2).\n\tThe $k[G]$-linearity is easy to check.\n\\end{proof}\n\\subsection*{Step IV}\nConsider the maps:\n\\begin{align*}\n\t\\Phi_0 : H^0(X, \\Omega_{X\/k}) \\oplus k[G]_B &\\to H^0(X, \\Omega_{X\/k}(R)) \\oplus I_{X\/Y}\\\\\n\t\\Phi_0(\\omega) &:= \\omega + \\res_G(\\omega) \\\\\n\t\\Phi_0 \\left(\\sum_{Q, g} a_{Q, g} g_Q \\right) &:= \\sum_{Q, g} a_{Q, g} \\cdot g^*(z) \\cdot \\eta_Q\n\\end{align*}\nand:\n\\begin{align*}\n\t\\Psi_0 : H^0(X, \\Omega_{X\/k}(R)) \\oplus I_{X\/Y} &\\to H^0(X, \\Omega_{X\/k}) \\oplus k[G]_B\\\\\n\t\\Psi_0(\\omega) &:= \\omega^{\\circ} + \\res_G(\\omega)\\\\\n\t\\Psi_0\\left(\\sum_{Q, g} a_{Q, g} g_Q \\right) &:=\n\t\\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q - \\sum_{Q, g} a_{Q, g} g_Q.\n\\end{align*}\nNote that $\\Phi_0$ and $\\Psi_0$ are well-defined $k[G]$-linear homomorphisms by Steps I--III.\nWe show now that $\\Phi_0$ and $\\Psi_0$ are mutually inverse.\nWe start by showing that $\\Phi_0 \\circ \\Psi_0 = \\id$.\nLet $\\omega \\in H^0(X, \\Omega_{X\/k})$. Then:\n\n\t\\begin{align*}\n\t\t\\Psi_0(\\Phi_0(\\omega)) &= \\Psi_0(\\omega + \\sum_{Q, g} \\res_Q(\\omega_g) g_Q)\\\\\n &= \\left(\\omega^{\\circ} + \\sum_{Q, g} \\res_Q(\\omega_g) g_Q \\right)\\\\\n &+ \\left(\\sum_{Q, g} \\res_Q(\\omega_g) g^*(z) \\eta_Q - \\sum_{Q, g} \\res_Q(\\omega_g) g_Q \\right)\\\\\n &= \\omega.\n\t\\end{align*}\n\n\tAnalogously, for $\\sum_{Q, g} a_{Q, g} g_Q \\in k[G]_B$:\n\n\t\\begin{align*}\n\t\t\\Psi_0\\left(\\Phi_0\\left(\\sum_{Q, g} a_{Q, g} g_Q\\right)\\right)\n\t\t&= \\Psi_0\\left(\\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q \\right)\\\\\n\t\t&= \\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q - \\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q\\\\\n\t\t&+ \\sum_{Q, g} a_{Q, g} g_Q\\\\\n\t\t&= \\sum_{Q, g} a_{Q, g} g_Q.\n\t\\end{align*}\n\n\t\n\tWe prove now that $\\Phi_0 \\circ \\Psi_0 = \\id$. If $\\omega \\in H^0(X, \\Omega_{X\/k}(R))$ then:\n\n\t\\begin{align*}\n\t\t\\Phi_0(\\Psi_0(\\omega)) &= \\Phi_0 \\left(\\omega^{\\circ} +\n\t\t\\sum_{Q, g} \\res_Q(\\omega_g) g_Q \\right).\n\t\\end{align*}\n\n\tRecall from Step III that $\\res_G(\\omega^{\\circ}) = 0$. Hence:\n\n\t\\begin{equation*}\n\t\t\\Phi_0(\\Psi_0(\\omega)) = \\omega^{\\circ} + \\sum_{Q, g} \\res_Q(\\omega_g) g^*(z) \\eta_Q\n\t\t= \\omega.\n\t\\end{equation*}\n\n\tFinally, for $\\sum_{Q, g} a_{Q, g} g_Q \\in I_{X\/Y}$:\n\n\t\\begin{align*}\n\t\t\\Phi_0\\left(\\Psi_0\\left(\\sum_{Q, g} a_{Q, g} g_Q\\right)\\right)\n\t\t&= \\Phi_0\\left(\\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q - \\sum_{Q, g} a_{Q, g} g_Q \\right)\\\\\n\t\t&= \\left(\\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q + \\sum_{Q, g} a_{Q, g} g_Q \\right)\\\\\n\t\t&- \\left(\\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q \\right)\\\\\n\t\t&= \\sum_{Q, g} a_{Q, g} g_Q.\n\t\\end{align*}\n\n\tThis ends the proof.\n\n\\section{Cohomology of the structure sheaf} \\label{sec:OX}\nThis section is dedicated to a proof of the following proposition. \n\\begin{Proposition} \\label{prop:regular_and_vanishing_functions}\n\tKeep assumptions of Theorem~\\ref{thm:main_thm}. \n\tThere exists the following isomorphism of $k[G]$-modules:\n\n\t\\[\n\tH^1(X, \\mc O_X) \\oplus k[G]^{\\# B - 1} \\cong H^1(X, \\mc O_X(-R)) \\oplus J_{X\/Y}.\n\t\\]\n\\end{Proposition}\nNote that this result might be also obtained by applying Serre duality to Proposition~\\ref{prop:log_diffs_and_diffs}. However, we prefer to give a direct proof, as we will need a description of the isomorphisms in the second part of the proof of Theorem~\\ref{thm:main_thm}.\\\\\n\nIn the sequel we will use an alternative description of sheaf cohomology of sheaves on a curve.\nBasically, it is a variation of \\v{C}ech cohomology for a cover consisting of\nan open set $U$ and ``infinitesimal neighbourhoods'' of points $Q \\not \\in U(k)$.\n\\begin{Lemma} \\label{lem:description_of_sheaf_cohomology}\n\tLet $Y$ be a smooth projective curve with the generic point~$\\eta$ over an algebraically closed field $k$. Let $S \\subset Y(k)$ be a finite non-empty set. Denote $U := Y \\setminus S$. Then for any locally free sheaf $\\mc F$ of finite rank on $Y$ we have a natural isomorphism:\n\n\t\\begin{align*}\n\t\tH^1(Y, \\mc F) &\\cong \\coker(\\mc F(U) \\to \\bigoplus_{Q \\in S} \\mc F_{\\eta}\/\\mc F_{Q}).\n\t\\end{align*}\n\n\\end{Lemma}\n\\begin{proof}\n\n\tLet $j : U \\hookrightarrow Y$ be the open immersion.\n\tIt is elementary to check that for any $Q \\in Y(k)$:\n\n\t\\[\n\tj_*(\\mc F|_U)_Q =\n\t\\begin{cases}\n\t\t\\mc F_Q, & \\textrm{ if } Q \\in U,\\\\\n\t\t\\mc F_{\\eta}, & \\textrm{ otherwise. }\n\t\\end{cases}\n\t\\]\n\n\tThis yields the exact sequence:\n\n\t\\begin{equation} \\label{eqn:F_to_FU_exact_sequence}\n\t\t0 \\to \\mc F \\to j_*(\\mc F|_U) \\to \\bigoplus_{Q \\in S} i_{Q, *}(\\mc F_{\\eta}\/\\mc F_Q) \\to 0,\n\t\\end{equation}\n\n\twhere $i_Q : \\Spec(\\mc O_{Y, Q}) \\to Y$ is the natural morphism.\n\tThe proof follows by taking the associated long exact sequence and noting that\n\t$H^1(Y, j_*(\\mc F|_U)) = H^1(U, \\mc F|_U) = 0$ by Serre's criterion on affineness (cf.~\\cite[Theorem~III.3.7]{Hartshorne1977}).\n\\end{proof}\nIn particular, the first cohomology group of any invertible sheaf on $Y$ may be identified with a quotient of $\\bigoplus_{Q \\in S} k(Y)$. In this context, the duality pairing\n\\[\n\\langle \\cdot, \\cdot \\rangle : H^0(Y, \\Omega_{Y\/k}(D)) \\times H^1(Y, \\mc O_Y(-D)) \\to k\n\\]\nis given by the formula:\n\\[\n\\langle \\omega, \\nu \\rangle := \\sum_{Q \\in S} \\res_Q(\\nu_Q \\cdot \\omega),\n\\]\nwhere the element $\\nu \\in H^1(Y, \\mc O_Y(D))$ is represented by\n$(\\nu_Q)_{Q \\in S} \\in \\bigoplus_{Q \\in S} k(Y)$.\nThe following lemma can be seen as a dual version of Lemma~\\ref{lem:main_lemma_Omega_Y}.\n\\begin{Lemma} \\label{lem:main_lemma_OX}\n\n\tKeep assumptions of Theorem~\\ref{thm:main_thm}. Let $Q \\in B$.\n\t\\begin{enumerate}[(1)]\n\t\t\\item $z_Q^{\\vee} := \\tr_{X\/X_Q}(z^{\\vee}) \\in \\mc O_{X_Q, Q}$.\n\t\t\n\t\t\\item Let $f \\in \\mc O_{X, Q}$. Then for every $g_1, g_2 \\in G$ satisfying\n\t\t$g_1 G_Q = g_2 G_Q$:\n\t\n\t\t\\[\n\t\tf_{g_1}(Q) = f_{g_2}(Q).\n\t\t\\]\n\t\n\t\t\n\t\t\\item Let $Q \\in B$ and $f \\in \\mc O_{X, Q}$. Then:\n\t\n\t\t\\[\n\t\t\tf \\in \\mc O_X(-R)_Q \\quad \\Leftrightarrow \\quad \\forall_{g \\in G} \\quad f_g(Q) = 0.\n\t\t\\]\n\t\n\t\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nBefore the proof note that the map:\n\\begin{equation} \\label{eqn:local_duality_X}\n\t(f, \\omega) \\mapsto \\sum_{P \\in \\pi^{-1}(Q)} \\res_P(f \\cdot \\omega)\n\t= \\res_Q(\\tr_{X\/Y}(f \\cdot \\omega))\n\\end{equation}\ninduces a duality between $k(X)\/\\mc O_{X, Q}(-D)$ and $\\Omega_{X, Q}(D)$ for any $D \\in \\Divv(X)$.\\\\\n\n(1) It suffices to show that $z_Q^{\\vee} \\in \\mc O_{X, Q}$. To this end note that for any $\\omega \\in \\Omega_{X\/k, Q}$:\n\n\t\\begin{align*}\n\t\t0 = \\sum_{g \\in G_Q} \\res_Q(\\omega_g)\n\t\t= \\sum_{g \\in G_Q} \\res_Q(\\tr_{X\/Y}(g^*(z^{\\vee}) \\cdot \\omega))\n\t\t= \\res_Q(\\tr_{X\/Y}(z^{\\vee}_Q \\cdot \\omega))\n\t\\end{align*}\n\n\tby Lemma~\\ref{lem:main_lemma_Omega_Y}~(2) and by~\\eqref{eqn:gth_component_trace_f}.\n\tThus $z_Q^{\\vee} \\in \\mc O_{X, Q}$ by the duality~\\eqref{eqn:local_duality_X} for $D = 0$.\\\\\n\t\n (2) Note that $f_{g_1}, f_{g_2} \\in \\mc O_{X, Q}$ by Lemma~\\ref{lem:inclusions_of_modules2}. Let $\\omega_Q \\in \\Omega_{k(Y)\/k}$ be any differential form with $\\ord_Q(\\omega_Q) = -1$, $\\res_Q(\\omega_Q) = 1$. By~\\eqref{eqn:residue_and_trace} and~\\eqref{eqn:gth_component_trace_f}:\n\n\t\\begin{align*}\n\t\tf_{g_1}(Q) &= \\res_Q(f_{g_1} \\cdot \\omega_Q)\\\\\n\t\t&= \\res_Q(\\tr_{X\/Y}(g_1^*(z) \\cdot f \\cdot \\omega_Q))\\\\\n\t\t&= \\sum_{P \\in \\pi^{-1}(Q)} \\res_P(f \\cdot g_1^*(z) \\cdot \\omega_Q)\n\t\\end{align*}\n\n\tand analogously $f_{g_2}(Q) = \\sum_{P \\in \\pi^{-1}(Q)} \\res_P(f \\cdot g_2^*(z) \\cdot \\omega_Q)$.\n\tNote that $g_1^*(z) \\omega_Q - g_2^*(z) \\omega_Q \\in \\Omega_{X\/k, Q}$ by Lemma~\\ref{lem:main_lemma_Omega_Y}~(2).\n\tHence:\n\n\t\\[\n\t\t(f_{g_1} - f_{g_2})(Q) = \\sum_{P \\in \\pi^{-1}(Q)} \\res_P(f \\cdot (g_1^*(z) \\omega_Q - g_2^*(z) \\omega_Q)) = 0.\n\t\\]\n\t\n (3) One implication is clear by Lemma~\\ref{lem:inclusions_of_modules2}. For the second implication, \n suppose that $f_g(Q) = 0$ for every $g \\in G$. Let $\\omega \\in \\Omega_{X\/k}(R)_Q$. Then\n\n \\[ \\omega' := \\omega - \\sum_{g \\in G} \\res_Q(\\omega_g) g^*(z) \\omega_Q \\in \\Omega_{X\/k, Q} \\]\n\n by Lemma~\\ref{lem:main_lemma_Omega_Y}~(2). Hence $f \\cdot \\omega'$ is regular at $Q$ and:\n\n \\begin{align*}\n \t\\res_Q(\\tr_{X\/Y}(f \\cdot \\omega)) &= \\res_Q(\\tr_{X\/Y}(f \\cdot \\omega')) + \\sum_{g \\in G} \\res_Q(\\omega_g) \\res_Q(\\tr_{X\/Y}(f \\cdot g^*(z) \\omega_Q))\\\\\n \t&= 0 + \\sum_{g \\in G} \\res_Q(\\omega_g) f_g(Q) = 0.\n \\end{align*}\nTherefore $f \\in \\mc O_X(-R)_Q$ by the duality~\\eqref{eqn:local_duality_X} for $D = R$.\n\\end{proof}\nWe discuss now the notation used in the proof of Proposition~\\ref{prop:regular_and_vanishing_functions}.\nBy Lemma~\\ref{lem:description_of_sheaf_cohomology} one can identify $H^1(Y, \\mc O_Y)$ and $H^1(Y, \\mc O_Y(-B))$ with certain quotients of\n$\\bigoplus_{Q \\in B} k(Y)$. Analogously, $H^1(X, \\mc O_X) \\cong H^1(Y, \\pi_*\\mc O_X)$ and $H^1(X, \\mc O_X(-R))$ are quotients of $\\bigoplus_{Q \\in B} k(X)$. By abuse of notation,\nwe often denote the element of the cohomology\ndetermined by an element of $\\bigoplus_B k(X)$ by the same letter.\nWe treat $\\bigoplus_{Q \\in B} k(X)$ as a $k(X)$-module.\nFor any $Q \\in B$ denote by $\\delta(Q)$ the element of $\\bigoplus_{B} k(X)$ with $1$ on the $Q$-th component and zero\non other components.\\\\\n\nFor any $\\nu \\in \\bigoplus_{Q \\in B} k(X)$, $Q \\in B$ and $g \\in G$\nwe adapt the following notation:\n\\begin{itemize}\n\t\\item $\\nu_Q \\in k(X)$ denotes the $Q$-th coordinate of $\\nu$,\n\t\n\t\\item $\\nu_{Q, g} \\in k(Y)$ are defined by the equality $\\nu_Q = \\sum_{g \\in G} g^*(z^{\\vee}) \\nu_{Q, g}$,\n\t\n\t\\item $\\nu_g \\in \\bigoplus_{B} k(Y)$ is the element with $\\nu_{Q, g}$ on the $Q$-th coordinate.\n\\end{itemize}\nConsider the following map for a fixed element $g \\in G$:\n\\begin{equation} \\label{eqn:projection_onto_gth_component}\n\tH^1(X, \\mc O_X(-R)) \\to H^1 \\left(Y, \\bigoplus_{g \\in G} g^*(z^{\\vee}) \\mc O_Y(-B) \\right)\n\t\\to H^1(Y, \\mc O_Y(-B)),\n\\end{equation}\nwhere:\n\\begin{itemize}\n\t\\item the first map is induced by the inclusion from Lemma~\\ref{lem:inclusions_of_modules2},\n\t\n\t\\item the second map is the projection onto the component corresponding to~$g$.\n\\end{itemize}\t\nNote that the image of an element of $H^1(X, \\mc O_X(-R))$ represented by $\\nu \\in \\bigoplus_B k(X)$ via the map~\\eqref{eqn:projection_onto_gth_component} is represented by $\\nu_g \\in \\bigoplus_B k(Y)$.\\\\\n\nWe identify $J_{G, H}$ with:\n\\[\nJ_{G, H} = k[G]\\bigg\/\\left(\\sum_{g \\in G} a_g \\cdot g : a_{g_1} = a_{g_2} \\textrm{ if } g_1 H = g_2 H \\right).\n\\]\nLet $k[G]_B^{\\vee}$ be the dual of the module $k[G]_B$. Note that $k[G]_B^{\\vee}$ may be identified with the module:\n\\[\n\\coker \\left(\\textrm{diag} : k[G] \\to \\bigoplus_B k[G] \\right).\n\\]\nand that $k[G]_B^{\\vee} \\cong k[G]^{\\# B - 1}$. For any $g \\in G$ and $Q \\in B$, denote the image of $g_Q$ in $J_{X\/Y}$ and in $k[G]_B^{\\vee}$ by $\\ol{g}_Q$.\n\n\\subsection*{Proof of Proposition~\\ref{prop:regular_and_vanishing_functions}}\n\n\tLet $\\eta_Q$ be the forms defined in Section~\\ref{sec:OmegaX} for a fixed $Q_0 \\in B$.\n\tAgain, we abbreviate $\\sum_{Q \\in B} \\sum_{g \\in G}$ to $\\sum_{Q, g}$.\n\tThe proof is divided into four steps.\n\n\t\\subsection*{Step I}\n\n\tThe map:\n\n\t\\[\n\t\\bigoplus_{Q \\in B} k(X) \\to \\bigoplus_{Q \\in B} k(X), \\quad \\nu \\mapsto \\nu^{\\circ} := \\nu - \\sum_{Q, g} \\langle \\eta_Q, \\nu_g \\rangle \\cdot g^*(z^{\\vee}) \\cdot \\delta(Q)\n\t\\]\n\n\tinduces a $k[G]$-linear map $H^1(X, \\mc O_X) \\to H^1(X, \\mc O_X(-R))$.\n\n\t\\begin{proof}[Proof of Step I]\n\t\tOne easily checks that this map is $k[G]$-linear.\n\t\tSuppose that $\\nu = (h)_Q$ for $h \\in \\mc O_X(V)$. Then $\\nu_g = (h_g)_Q$ and\n\t\t$h_g \\in \\mc O_Y(U)$ by Lemma~\\ref{lem:inclusions_of_modules2}.\n\t\tIt follows that $\\langle \\eta_Q, \\nu_g \\rangle = 0$ by residue theorem.\n\t\tTherefore $\\nu^{\\circ}$ is trivial in $H^1(X, \\mc O_X(-R))$.\n\t\t\n\t\tConsider now an element $\\nu \\in \\bigoplus_B k(X)$ satisfying $\\nu_Q \\in \\mc O_{X, Q}$\n\t\tfor every $Q \\in B$. Note that then $\\nu_{Q, g} \\in \\mc O_{Y, Q}$ by Lemma~\\ref{lem:inclusions_of_modules2} and:\n\t\n\t\t\\[\n\t\t\\langle \\eta_Q, \\nu_g \\rangle = \\nu_{Q, g}(Q) - \\nu_{Q_0, g}(Q_0).\n\t\t\\]\n\t\n\t\tLemma~\\ref{lem:main_lemma_OX}~(3) implies that for any $Q \\in B$:\n\t\n\t\t\\[\n\t\t\\nu_Q - \\sum_{g \\in G} g^*(z^{\\vee}) \\cdot \\nu_{Q, g}(Q)\n\t\t= \\sum_{g \\in G} g^*(z^{\\vee}) \\cdot (\\nu_{Q, g} - \\nu_{Q, g}(Q))  \\in \\mc O_X(-R)_Q,\n\t\t\\]\n\t\n\t\tsince $\\nu_{Q, g} - \\nu_{Q, g}(Q) \\in \\mf m_{Y, Q}$.\n\t\tThus in $H^1(X, \\mc O_X)$:\n\t\n\t\t\\begin{align*}\n\t\t\t\\nu^{\\circ} &= \\nu - \\left(\\sum_{g \\in G} (\\nu_{Q, g}(Q) - \\nu_{Q_0, g}(Q_0)) \\cdot g^*(z^{\\vee}) \\right)_Q\\\\\n\t\t\t&= \\left(\\sum_{g \\in G} \\nu_{Q_0, g}(Q_0) \\cdot g^*(z^{\\vee}) \\right)_{Q} = 0,\n\t\t\\end{align*}\n\t\n\t\tsince $\\sum_{g \\in G} \\nu_{Q_0, g}(Q_0) \\cdot g^*(z^{\\vee}) \\in \\mc O_X(V)$. The statement follows.\n\t\\end{proof}\n\n\t\\subsection*{Step II}\n\n\tThe map:\n\n\t\\[\n\te_G : \\bigoplus_{Q \\in B} k(X) \\to \\bigoplus_{Q \\in B} k[G], \\qquad \n\t\\nu \\mapsto \\sum_{Q, g} \\langle \\eta_Q, \\nu_g \\rangle \\cdot g_Q\n\t\\]\n\n\tinduces well-defined $k[G]$-linear maps\n\n\t\\[\n\tH^1(X, \\mc O_X) \\to J_{X\/Y} \\textrm{ and } H^1(X, \\mc O_X(-R)) \\to k[G]_B^{\\vee}.\n\t\\]\n\n\t\\begin{proof}[Proof of Step II]\n\t\tSuppose that $\\nu = (h)_Q$ for $h \\in \\mc O_X(V)$. Then $\\langle \\eta_Q, \\nu_g \\rangle = 0$ as proven in Step I.\n\t\tHence $\\nu$ maps to $0$ in $J_{X\/Y}$. Assume now that an element $\\nu \\in \\bigoplus_B k(X)$ satisfies $\\nu_Q \\in \\mc O_{X, Q}$\n\t\tfor every $Q \\in B$. Note that then:\n\t\n\t\t\\[\n\t\t\\langle \\eta_Q, \\nu_g \\rangle = \\nu_{Q, g}(Q) - \\nu_{Q_0, g}(Q_0).\n\t\t\\]\n\t\n\t\tBut $\\sum_{Q, g} \\nu_g(Q) \\ol g_Q = 0$ in $J_{X\/Y}$ by Lemma~\\ref{lem:main_lemma_OX}~(2) and\n\t\tthe definition of $J_{X\/Y}$. Moreover, $\\sum_{Q, g} \\nu_g(Q_0) \\ol g_Q = 0$ in\n\t\t$J_{X\/Y}$ (since this is the image of the element $\\sum_{g \\in G} \\nu_g(Q_0) \\cdot g \\in k[G]$).\n\t\tHence $\\nu$ also maps to $0$ in $J_{X\/Y}$. This shows that the map $H^1(X, \\mc O_X) \\to J_{X\/Y}$\n\t\tis well-defined. One checks that the map $H^1(X, \\mc O_X(-R)) \\to k[G]_B^{\\vee}$ is well-defined in a similar\n\t\tmanner.\n\t\\end{proof}\n\n\t\\subsection*{Step III}\n\n\tThe map:\n\n\t\\begin{equation} \\label{eqn:map_kG_kX}\n\t\t\\bigoplus_{Q \\in B} k[G] \\to \\bigoplus_{Q \\in B} k(X), \\quad\n\t\tg_Q \\to g^*(z^{\\vee}) \\cdot \\delta(Q)\n\t\\end{equation}\n\n\tinduces well-defined $k[G]$-linear maps\n\n\t\\[\n\t\tJ_{X\/Y} \\to H^1(X, \\mc O_X) \\quad \\textrm{ and } \\quad k[G]_B^{\\vee} \\to H^1(X, \\mc O_X(-R)).\n\t\\]\n\n\t\\begin{proof}[Proof of Step III]\n\t\tWe show that the homomorphism $J_{X\/Y} \\to H^1(X, \\mc O_X)$ is well-defined; the proof of the\n\t\tsecond statement is analogous. Consider an element of $\\bigoplus_B k[G]$ of the form\n\t\t$\\sum_{g \\in g_0 G_Q} g_Q$ for some $g_0 \\in G$, $Q \\in B$.\n\t\tIts image through the map~\\eqref{eqn:map_kG_kX} is\n\t\t$g_0^*(z_Q^{\\vee}) \\cdot \\delta(Q)$.\n\t\tHowever, $z_Q^{\\vee} \\in \\mc O_{X, Q}$ by Lemma~\\ref{lem:main_lemma_OX}~(1) and thus\n\t\t$g_0^*(z_Q^{\\vee}) \\cdot \\delta(Q) = 0$ in $H^1(X, \\mc O_X)$.\n\n\t\tFinally, observe that the image of an element $\\sum_{Q, g} a_g g_Q \\in \\bigoplus_B k[G]$ through the map~\\eqref{eqn:map_kG_kX} is $\\sum_{Q, g} a_g \\cdot (g^*(z^{\\vee}))_Q$.\n\t\tThis is trivial in $H^1(X, \\mc O_X)$, since $g^*(z^{\\vee}) \\in \\mc O_X(-R)(V)$.\n\t\t\\end{proof}\n\n\t\\subsection*{Step IV}\n\tConsider the following two maps:\n\n\t\\begin{align*}\n\t\\Phi_1 : H^1(X, \\mc O_X) \\oplus k[G]_B^{\\vee} &\\to H^1(X, \\mc O_X(-R)) \\oplus J_{X\/Y},\\\\\n\t\\Phi_1(\\nu) &:= \\nu^{\\circ} + e_G(\\nu), \\\\\n\t\\Phi_1 (\\ol g_Q) &:= g^*(z^{\\vee}) \\cdot \\delta(Q) - \\ol g_Q,\n\t\\end{align*}\n\n\n\t\\begin{align*}\n\t\\Psi_1 : H^1(X, \\mc O_X(-R)) \\oplus J_{X\/Y} &\\to H^1(X, \\mc O_X) \\oplus k[G]_B^{\\vee},\\\\\n\t\\Psi_1(\\nu) &:= \\nu + e_G(\\nu), \\\\\n\t\\Psi_1(\\ol g_Q) &:= g^*(z^{\\vee}) \\cdot \\delta(Q).\n\t\\end{align*}\t\n\n\tSteps I--III imply that $\\Phi_1$ and $\\Psi_1$ are well-defined $k[G]$-linear homomorphisms.\n\tOne easily checks that they are mutually inverse.\n\n\\section{The de Rham cohomology} \\label{sec:dR}\nIn this section we prove the part of Theorem~\\ref{thm:main_thm} concerning the de Rham cohomology.\nTo this end we need to compare $H^1_{dR}(X\/k)$ with the following variant of the logarithmic de Rham cohomology.\nLet for any finite set $S \\subset Y(k)$, $H^1_{dR, S}(Y\/k)$ denote the hypercohomology of the complex:\n\\[\n\\Omega_{Y\/k}^{\\bullet}(\\pm S) : (\\mc O_{Y\/k}(-S) \\stackrel{d}{\\longrightarrow} \\Omega_{Y\/k}(S)).\n\\]\nNote that there is a ``Hodge--de Rham exact sequence'' for $H^1_{dR, S}(Y\/k)$:\n\\begin{equation} \\label{eqn:hdr_exact_sequence_for_log_de_rham}\n\t0 \\to H^0(Y, \\Omega_{Y\/k}(S)) \\to H^1_{dR, S}(Y\/k) \\to H^1(Y, \\mc O_Y(-S)) \\to 0.\n\\end{equation}\nThe following result is an analogue of Propositions~\\ref{prop:log_diffs_and_diffs} and~\\ref{prop:regular_and_vanishing_functions}.\n\\begin{Proposition} \\label{prop:normal_and_log_de_rham}\n\tThere exists an isomorphism between the $k[G]$-modules:\n\n\t\\begin{equation*}\n\t\tM_1 := H^1_{dR}(X\/k) \\oplus k[G]_B \\oplus k[G]_B^{\\vee}\n\t\\end{equation*}\n\n\tand\n\n\t\\begin{equation*}\n\t\tM_2 := H^1_{dR, R}(X\/k) \\oplus I_{X\/Y} \\oplus J_{X\/Y}.\n\t\\end{equation*}\n\n\\end{Proposition}\nWe give now a description of sheaf hypercohomology, that generalizes Lemma~\\ref{lem:description_of_sheaf_cohomology}.\n\\begin{Lemma}\n\tKeep the setup of Lemma~\\ref{lem:description_of_sheaf_cohomology}. Let $\\mc F^{\\bullet} = (\\mc F^0 \\stackrel{d}{\\rightarrow} \\mc F^1)$\n\tbe a cochain complex of locally free $\\mc O_Y$-modules of finite rank with a $k$-linear differential.\n\tThen we have a natural isomorphism:\n\n\t\\begin{align*}\n\t\t\\HH^1(Y, \\mc F^{\\bullet}) &\\cong Z^1_S(\\mc F^{\\bullet})\/B^1_S(\\mc F^{\\bullet}),\n\t\\end{align*}\n\n\twhere:\n\t\\begin{align*}\n\t\tZ^1_{S}(\\mc F^{\\bullet}) &:= \\{ (\\omega, (h_Q)_{Q \\in S} ) : \\omega \\in \\mc F^0(U), \n\t\th_Q \\in \\mc F^1_{\\eta}, \\, \\omega - d h_Q \\in \\mc F^0_Q \\},\\\\\n\t\tB^1_{S}(\\mc F^{\\bullet}) &:= \\{ (dh, (h + h_Q)_{Q \\in S} ) : h \\in \\mc F^1(U), \\,\n\t\th_Q \\in \\mc F^1_Q \\}.\n\t\\end{align*}\n\n\tMoreover, the maps\n\n\t\\begin{align*}\n\t\tH^0(Y, \\mc F^0) \\to \\HH^1(Y, \\mc F^{\\bullet}) \\quad \\textrm{ and } \\quad \n\t\t\\HH^1(Y, \\mc F^{\\bullet}) \\to H^1(Y, \\mc F^1)\n\t\\end{align*}\n\n\tare induced by the maps:\n\n\t\\[\n\t\t\\begin{array}{ccc}\n\t\t\tH^0(Y, \\mc F^0) &\\to& Z^1_S(\\mc F^{\\bullet})\\\\\n\t\t\t\\omega &\\mapsto& (\\omega, (0)_{Q \\in S})\n\t\t\\end{array}\n\t\t\\qquad \\textrm{ and } \\qquad\n\t\t\\begin{array}{ccc}\n\t\t\tZ^1_S(\\mc F^{\\bullet}) &\\to& \\bigoplus_{Q \\in S} \\mc F^1_{\\eta}\\\\\n\t\t\t(\\omega, (h_Q)_{Q \\in S} ) &\\mapsto& (h_Q)_{Q \\in S}\n\t\t\\end{array}\n\t\\]\n\n\trespectively.\n\\end{Lemma}\n\\begin{proof}\n\n\tThe exact sequence~\\eqref{eqn:F_to_FU_exact_sequence} easily implies that $\\mc F^{\\bullet}$ is the kernel of the map of complexes:\n\n\t\\[\n\tj_*(\\mc F^{\\bullet}|_U) \\to \\bigoplus_{Q \\in S} i_{Q, *}(\\mc F^{\\bullet}_{\\eta}\/\\mc F^{\\bullet}_Q).\n\t\\]\n\n\tLet $C^{\\bullet}$ be the mapping cone of this map, i.e. the total complex of the double complex:\n\n\t\\begin{center}\n\t\n\t\t\\begin{tikzcd}\n\t\t\tj_*(\\mc F^0|_U) \\arrow[r] \\arrow[d] & {\\bigoplus_{Q \\in S} i_{Q, *}(\\mc F^0_{\\eta}\/\\mc F^0_Q)} \\arrow[d] \\\\\n\t\t\tj_*(\\mc F^1|_U) \\arrow[r]           & {\\bigoplus_{Q \\in S} i_{Q, *}(\\mc F^1_{\\eta}\/\\mc F^1_Q).}\n\t\t\\end{tikzcd}\n\t\\end{center}\n\n\tThen $\\mc F^{\\bullet}$ is isomorphic to $C^{\\bullet}[-1]$ in the derived category. In particular, $\\HH^1(Y, \\mc F^{\\bullet}) \\cong \\HH^0(Y, C^{\\bullet})$. It is immediate from the definition that\n\n\t\\[\n\t\\HH^0(Y, C^{\\bullet}) \\cong Z^1_S(\\mc F^{\\bullet})\/B^1_S(\\mc F^{\\bullet}).\n\t\\]\n\n\tThe second statement follows from the functoriality for the maps $\\mc F^0[0] \\to \\mc F^{\\bullet}$\n\tand $\\mc F^{\\bullet} \\to \\mc F^1[1]$.\n\\end{proof}\nIn order to prove Proposition~\\ref{prop:normal_and_log_de_rham} we lift the map\n$J_{X\/Y} \\to H^1(X, \\mc O_X)$\nfrom Step III of the proof of Proposition~\\ref{prop:regular_and_vanishing_functions}\nto a map $J_{X\/Y} \\to H^1_{dR}(X\/k)$. To this end we need to decompose the differential form $dz^{\\vee}$ into certain ``local components''. This is achieved by the following lemma.\n\\begin{Lemma} \\label{lem:xiQ_existence}\n\tThere exists a system of differential forms $(\\xi_Q)_{Q \\in B}$, $\\xi_Q \\in \\Omega_{k(X)\/k}$\n\tsuch that the following conditions are satisfied:\n\n\t\\begin{enumerate}[(1)]\n\t\t\\item $\\xi_Q \\equiv dz^{\\vee} \\pmod{\\Omega_{X, Q}}$,\n\t\t\\item $\\xi_Q \\in \\Omega_{X, Q'}$ for every $Q' \\in Y(k)$, $Q' \\neq Q$,\n\t\t\\item $\\sum_{g \\in g_0 G_Q} g^*(\\xi_Q) = 0$ for every $g_0 \\in G$,\n\t\t\\item $\\sum_{Q \\in B} \\xi_Q = dz^{\\vee}$.\n\t\\end{enumerate}\n\n\\end{Lemma}\n\\begin{proof}\n\n\tDenote $J' := \\bigoplus_{Q \\in B} J_{G, G_Q}$ and $d_g := \\sum_{Q \\in B} \\ol g_Q \\in J'$ for any $g \\in G$.\n\tThen $\\sum_{g \\in G} d_g = 0$. \n\tNote that the elements $(d_g)_{g \\neq e}$ are linearly independent over $k$.\n\tIndeed, suppose to the contrary that $\\sum_{g \\in G} a_g \\cdot d_g = 0$ in $J'$ for some\n\t$a_g \\in k$, where $a_e := 0$. Then by the definition of $J'$ we have $a_{g_1} = a_{g_2}$ for any $g_1, g_2 \\in G$ satisfying $g_1 G_Q = g_2 G_Q$ for some $Q \\in B$. But by Lemma~\\ref{lem:GME_implies_no_etale_cover}~(2)\n\t$G = \\langle G_Q : Q \\in B \\rangle$. Hence any $g \\in G$ can be written in the\n\tform $g_1 \\cdot \\ldots \\cdot g_m$ for $g_i \\in G_{Q_i}$,\n\t$Q_i \\in B$. Therefore:\n\n\t\\[\n\t\t0 = a_e = a_{g_1} = a_{g_1 g_2} = \\ldots = a_g.\n\t\\]\n\tHence $(d_g)_{g \\neq e}$ are linearly independent. Observe now that\n\n\t\\[\n\tJ' = \\Span_k(\\{ \\ol g_Q : g \\in G, Q \\in B \\} \\cup \\{ d_g : g \\neq e \\}).\n\t\\]\n\n\tHence there exists a $k$-basis of $J'$ of the form $\\mc B = \\mc B' \\cup \\{ d_g : g \\neq e \\}$, where\n\t$\\mc B' \\subset \\{ \\ol g_Q : g \\in G, Q \\in B \\}$.\n\tWe define now two $k$-linear homomorphisms, using the basis $\\mc B$.\\\\\n\t\n\tLet $V := \\ker \\left( \\sum : \\bigoplus_{Q \\in B} k \\to k \\right)$.\n\tConsider the $k$-linear homomorphism $r = (r_Q)_Q : J' \\to V$ defined by its values on $\\mc B$:\n\n\t\\begin{itemize}\n\t\t\\item $r(d_g) = (\\res_Q((dz^{\\vee})_g))_Q$ for any $g \\neq e$,\n\t\t\\item $r(\\ol{g}_Q) = (0)_Q$ for $\\ol g_Q \\in \\mc B'$.\n\t\\end{itemize}\n\n\tWe define also the $k$-linear homomorphism\n\n\t\\[\n\t\\varphi : J' \\to \\Omega_{k(Y)\/k},\n\t\\]\n\n\tby its values on $\\mc B$:\n\n\t\\begin{itemize}\n\t\t\\item $\\varphi(d_g) = (dz^{\\vee})_g$ for $g \\in G$, $g \\neq e$,\n\t\t\\item for any $\\ol g_Q \\in \\mc B'$, $\\varphi(\\ol g_Q) \\in \\Omega_{k(Y)\/k}$ is any differential form\n\t\tthat is regular on $Y \\setminus B$ and satisfies:\n\t\n\t\t\\begin{align*}\n\t\t\t\\varphi(\\ol g_Q) &\\equiv (dz^{\\vee})_g \\pmod{\\Omega_{Y\/k}(B)_Q},\\\\\n\t\t\t\\varphi(\\ol g_Q) &\\equiv 0 \\pmod{\\Omega_{Y\/k}(B)_{Q'}} \\quad \\textrm{ for } Q' \\in B, Q' \\neq Q,\\\\\n\t\t\t\\res_{Q'}(\\varphi(\\ol g_Q)) &= r_{Q'}(\\ol g_Q) \\quad \\textrm{ for } Q' \\in B\n\t\t\\end{align*}\n\t\n\t\t(observe that such a form exists by~\\eqref{eqn:constructing_diff_forms}, since $\\sum_{Q'} r_{Q'}(\\ol g_Q) = 0$).\n\t\\end{itemize}\n\n\tNote that then:\n\n\t\\begin{itemize}\n\t\t\\item $\\res_Q(\\varphi(\\alpha)) = r_Q(\\alpha)$ for any $\\alpha \\in J'$ (since this equality holds for $\\alpha \\in \\mc B$),\n\t\t\n\t\t\\item $\\varphi(d_e) = (dz^{\\vee})_e$, since $d_e = - \\sum_{g \\neq e} d_g$ and\n\t\tby~\\eqref{eqn:trace_of_dual_z}:\n\t\n\t\t\\[\n\t\t\\sum_{g \\in G} (dz^{\\vee})_g = \\tr_{X\/Y}(dz^{\\vee}) = d \\tr_{X\/Y}(z^{\\vee}) = 0.\n\t\t\\]\n\t\n\t\\end{itemize}\n\n\tDefine for any $Q \\in B$:\n\n\t\\[\n\t\\xi_Q := \\sum_{g \\in G} g^*(z) \\cdot \\varphi(\\ol g_Q).\n\t\\]\n\n\tWe check now that the forms~$\\xi_Q$ satisfy the listed conditions. \n\tNote that for any $g \\in G$:\n\n\t\\[\n\t(dz^{\\vee})_g = \\varphi(d_g) = \\varphi \\left(\\sum_{Q \\in B} \\ol g_Q \\right) = \\sum_{Q \\in B} \\xi_{Q, g}.\n\t\\]\n\n\tHence:\n\n\t\\begin{align*}\n\t\t\\sum_{Q \\in B} \\xi_Q &= \\sum_{Q, g} g^*(z) \\xi_{Q, g}\n\t\t= \\sum_{g \\in G} g^*(z) \\sum_{Q \\in B} \\xi_{Q, g}\\\\\n\t\t&= \\sum_{g \\in G} g^*(z) (dz^{\\vee})_g = dz^{\\vee},\n\t\\end{align*}\n\n\twhich proves~(4). Moreover for any $g_0 \\in G$, $Q \\in B$:\n\n\t\\begin{align*}\n\t\t0 &= \\varphi \\left(\\sum_{g \\in g_0 G_Q} \\ol g_Q \\right) = \\sum_{g \\in g_0 G_Q} \\xi_{Q, g}.\n\t\\end{align*}\n\n\tHence, if $G\/G_Q = \\{ g_1 G_Q, \\ldots, g_r G_Q \\}$:\n\n\t\\begin{align*}\n\t\t\\sum_{g \\in g_0 G_Q} g^*(\\xi_Q) = \\sum_{i = 1}^r (g_i \\cdot g_0)^*(z_Q) \\sum_{g \\in g_i G_Q} \\xi_{Q, g} = 0\n\t\\end{align*}\n\n\tand~(3) is also true.\\\\\n\tWe prove now the properties (1) and~(2). Let $\\varphi_1$ denote the composition\n\n\t\\[\n\tJ' \\stackrel{\\varphi}{\\longrightarrow} \\Omega_{k(Y)\/k} \\to \\bigoplus_{Q \\in B} \\Omega_{k(Y)\/k}\/\\Omega_{Y\/k}(B)_Q.\n\t\\]\n\n\tConsider the $k$-linear homomorphism:\n\n\t\\begin{align*}\n\t\t\\wt{\\varphi}_2 : \\bigoplus_{Q \\in B} k[G]  &\\to \\bigoplus_{Q \\in B} \\Omega_{k(Y)\/k}\/\\Omega_{Y\/k}(B)_Q,\\\\\n\t\tg_Q &\\mapsto (dz^{\\vee})_g \\cdot \\delta(Q).\n\t\\end{align*}\n\n\tNote that $\\wt{\\varphi}_2$ induces a map:\n\n\t\\[\n\t\\varphi_2 : J' \\to \\bigoplus_{Q \\in B} \\Omega_{k(Y)\/k}\/\\Omega_{Y\/k}(B)_Q.\n\t\\]\n\n\tIndeed, for any $g_0 \\in G$, $Q \\in B$:\n\n\t\\begin{equation} \\label{eqn:chi_is_regular}\n\t\t\\sum_{g \\in g_0 G_Q} (dz^{\\vee})_g \\in \\Omega_{X_Q\/k, Q} \\cap \\Omega_{k(Y)\/k} = \\Omega_{Y\/k, Q},\n\t\\end{equation}\n\n\tsince $\\sum_{g \\in g_0 G_Q} (dz^{\\vee})_g$ may be written as a combination of forms $h^*(dz_Q^{\\vee})$ for $h \\in H$ with coefficients in $\\mc O_{X_Q, Q}$ (cf. Lemma~\\ref{lem:main_lemma_Omega_Y}~(1) and Lemma~\\ref{lem:main_lemma_OX}~(1)). Therefore:\n\n\t\\begin{align*}\n\t\t\\wt{\\varphi}_2 \\left(\\sum_{g \\in g_0 G_Q} g_Q \\right) &= \\sum_{g \\in g_0 G_Q} (dz^{\\vee})_g \\cdot \\delta(Q)\\\\\n\t\t&= 0 \\qquad \\textrm{ in } \\bigoplus_{Q \\in B} \\Omega_{k(Y)\/k}\/\\Omega_{Y\/k}(B)_Q\n\t\\end{align*}\n\n\tand $\\varphi_2$ is well-defined. Observe now that for any $g \\in G$:\n\n\t\\begin{align*}\n\t\t\\varphi_2(d_g) &= \\varphi_2 \\left(\\sum_{Q \\in B} \\ol g_Q \\right)\n\t\t= \\sum_{Q \\in B} (dz^{\\vee})_g \\cdot \\delta(Q)\\\\\n\t\t&= \\left((dz^{\\vee})_g \\right)_{Q \\in B} \\qquad \\textrm{ in } \\bigoplus_{Q \\in B} \\Omega_{k(Y)\/k}\/\\Omega_{Y\/k}(B)_Q.\n\t\\end{align*}\n\n\tTherefore $\\varphi_1$ and $\\varphi_2$ agree on $\\mc B$. Hence $\\varphi_1 = \\varphi_2$\n\tand for every $Q \\in B$, $g \\in G$:\n\n\t\\begin{align*}\n\t\t\\varphi(\\ol g_Q) &\\equiv (dz^{\\vee})_g \\pmod{\\Omega_{Y\/k}(B)_Q}\\\\\n\t\t\\varphi(\\ol g_Q) &\\equiv 0 \\pmod{\\Omega_{Y\/k}(B)_{Q'}} \\qquad \\textrm{ for } Q' \\in B, Q' \\neq Q.\n\t\\end{align*}\n\n\tUsing Lemma~\\ref{lem:inclusions_of_modules} one easily deduces that $\\xi_Q \\in \\Omega_{X\/k}(R)_{Q'}$\n\tfor $Q' \\in B$, $Q' \\neq Q$ and $\\xi_Q - dz^{\\vee} \\in \\Omega_{X\/k}(R)_Q$. Finally, for any\n\t$g_0 \\in G$ and $Q' \\in B$, $Q' \\neq Q$:\n\n\t\\begin{align*}\n\t\t\\sum_{g \\in g_0 G_{Q'}} \\res_{Q'}(\\xi_{Q, g}) &= \\sum_{g \\in g_0 G_{Q'}} r_{Q'}(\\ol g_Q)\n\t\t= r_{Q'} \\left(\\sum_{g \\in g_0 G_{Q'}} \\ol g_Q \\right)\\\\\n\t\t&= r_{Q'}(0) = 0.\n\t\\end{align*}\n\n\tTherefore (2) holds by Lemma~\\ref{lem:main_lemma_Omega_Y}~(2). Analogously,\n\tfor any $Q \\in B$, $g_0 \\in G$, using~\\eqref{eqn:chi_is_regular}:\n\n\t\\begin{align*}\n\t\t\\sum_{g \\in g_0 G_{Q}} \\res_Q(\\xi_{Q, g} - (dz^{\\vee})_g)\n\t\t&= r_Q \\left(\\sum_{g \\in g_0 G_{Q}} \\ol g_Q \\right) - \\res_Q \\left(\\sum_{g \\in g_0 G_Q} (dz^{\\vee})_g \\right)\\\\\n\t\t&= r_Q(0) - 0 = 0.\n\t\\end{align*}\n\n\tThus $\\xi_Q - dz^{\\vee} \\in \\Omega_{X\/k, Q}$ by Lemma~\\ref{lem:main_lemma_Omega_Y}~(2), which proves~(1).\n\\end{proof}\nIn the proof of Proposition~\\ref{prop:normal_and_log_de_rham} we will consider two modules\nisomorphic to $\\bigoplus_B k[G]$:\n\\begin{itemize}\n\t\\item the first one contains $I_{X\/Y}$ and $k[G]_B$,\n\t\n\t\\item the second one surjects onto $J_{X\/Y}$ and $k[G]_B^{\\vee}$.\n\\end{itemize}\nIn order to distinguish the elements of those two modules, we overline the elements of the latter module, e.g.\n$\\sum_{Q, g} a_{Q g} \\ol{g}_Q$. \n\\begin{proof}[Proof of Proposition~\\ref{prop:normal_and_log_de_rham}]\nFix a point $Q_0 \\in B$ and let $\\eta_Q$ be as defined in Section~\\ref{sec:OmegaX}.\n\\subsection*{Step I}\nFor any $Q \\in B$ the element\n\\[\n\\Xi_Q := (\\xi_Q, z^{\\vee} \\cdot \\delta(Q)) \\in \\Omega_{X\/k}(V) \\times \\bigoplus_B k(X)\n\\]\nbelongs to $Z^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet})$. \nMoreover, the map:\n\\[\n\\bigoplus_{Q \\in B} k[G] \\to Z^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet}), \\qquad g_Q \\mapsto g^*(\\Xi_Q)\n\\]\ninduces a $k[G]$-linear map $J_{X\/Y} \\to H^1_{dR}(X\/k)$.\n\\begin{proof}[Proof of Step I]\n\tThe first statement follows immediately by Lemma~\\ref{lem:xiQ_existence}~(1) and~(2). For the second claim, note that by Lemma~\\ref{lem:xiQ_existence}(3) and Lemma~\\ref{lem:main_lemma_OX}~(1) for any $g_0 \\in G$, $Q \\in B$:\n\n\t\\begin{align*}\n\t\t\\sum_{g \\in g_0 G_Q} g^*((\\xi_Q, z^{\\vee} \\cdot \\delta(Q))) \n\t\t&= (0, g_0^*(z_Q^{\\vee}) \\cdot \\delta(Q)) \\in B^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet}).\n\t\\end{align*}\n\n\tFinally, by Lemma~\\ref{lem:xiQ_existence}~(4):\n\n\t\\begin{align*}\n\t\t\\sum_{Q \\in B} g^*(\\Xi_Q) &= (d g^*(z^{\\vee}), \\sum_{Q \\in B} g^*(z^{\\vee}) \\delta(Q))\\\\\n\t\t&= (d g^*(z^{\\vee}), (g^*(z^{\\vee}))_{Q \\in B}) \\in B^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet}).\n\t\\end{align*}\n\n\tHence in $H^1_{dR}(X\/k)$:\n\n\t\\begin{align*}\n\t\t\\forall_{g_0 \\in G, Q \\in B} \\, \\sum_{g \\in g_0 G_Q} g^*(\\Xi_Q) = 0 \\quad \\textrm{ and } \\quad \\forall_{g \n\t\t\\in G}\n\t\t\\sum_{Q \\in B} g^*(\\Xi_Q) = 0,\n\t\\end{align*}\n\n\twhich proves the claim.\n\\end{proof}\n\\subsection*{Step II}\nThe map:\n\\[\n(\\omega, \\nu) \\mapsto (\\omega, \\nu)^{\\circ} := (\\omega, \\nu) - \\sum_{Q, g} \\langle \\eta_Q, \\nu_g \\rangle \\cdot g^*(\\Xi_Q)\n\\]\ndefines a $k[G]$-linear homomorphism $H^1_{dR}(X\/k) \\to H^1_{dR, R}(X\/k)$. This map lifts the map $H^1(X, \\mc O_X) \\to H^1(X, \\mc O_X(-R))$ from Step I of the proof\nof Proposition~\\ref{prop:regular_and_vanishing_functions}.\n\\begin{proof}[Proof of Step II]\nIf $(\\omega, \\nu) \\in Z^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet})$, then\nclearly $(\\omega, \\nu)^{\\circ} \\in Z^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet}(\\pm R))$.\nSuppose now that $(\\omega, \\nu) = (dh, (h)_Q)$ for $h \\in \\mc O_X(V)$. Then\n\\[ (\\omega, \\nu)^{\\circ} = (\\omega, \\nu) \\in B^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet}(\\pm R)),\\]\nsince by the residue theorem $\\langle \\eta_Q, \\nu_g\\rangle = 0$ for every $Q, g$.\nAssume now that $\\nu \\in \\bigoplus_{Q \\in B} \\mc O_{X, Q}$. Then $\\langle \\eta_Q, \\nu_g \\rangle = \\nu_{Q, g}(Q) - \\nu_{Q_0, g}(Q_0)$.\nNote that $\\sum_{Q, g} \\nu_{Q, g}(Q) \\cdot g^*(\\xi_Q) = 0$ by Lemma~\\ref{lem:main_lemma_OX}~(2) and Lemma~\\ref{lem:xiQ_existence}~(3).\nMoreover, $\\sum_{Q, g} \\nu_{Q_0, g}(Q_0) \\cdot g^*(\\xi_Q) = \\sum_{g} \\nu_{Q_0, g}(Q_0) dg^*(z^{\\vee})$ by Lemma~\\ref{lem:xiQ_existence}~(4). \nTherefore:\n\\begin{align*}\n\t(0, \\nu)^{\\circ} &= \\bigg(0, \\nu - \\sum_{Q, g} \\nu_{Q, g}(Q) \\cdot g^*(z^{\\vee}) \\cdot \\delta(Q) \\bigg) + \\sum_{g} \\nu_{Q_0, g}(Q_0) \\cdot \\bigg(dg^*(z^{\\vee}), (g^*(z^{\\vee}))_Q \\bigg).\n\\end{align*}\nBut $\\nu - \\sum_{Q, g} \\nu_{Q, g}(Q) \\cdot g^*(z^{\\vee}) \\cdot \\delta(Q) \\in \\mc O_X(-R)_{Q'}$ for every $Q' \\in B$\nby Lemma~\\ref{lem:main_lemma_OX}~(3). Hence $(0, \\nu)^{\\circ} \\in B^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet}(\\pm R))$.\nThis proves that the map $H^1_{dR}(X\/k) \\to H^1_{dR, R}(X\/k)$ is well-defined. The second statement\nis immediate.\n\\end{proof}\n\n\\subsection*{Step III}\nThe map:\n\\begin{align*}\n\tZ^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet}) &\\to \\bigoplus_{Q \\in B} k[G],\\\\\n\t(\\omega, \\nu) &\\mapsto \\sum_{Q, g} (\\res_Q((\\omega - d \\nu_Q)_g) - \\langle g^*(\\xi_Q), \\nu \\rangle) \\cdot g_Q\n\\end{align*}\ninduces a map $H^1_{dR}(X\/k) \\to I_{X\/Y}$.\n\\begin{proof}[Proof of Step III]\nWe start by checking that image of this map lies in $I_{X\/Y}$.\nIndeed, $\\omega - d \\nu_Q \\in \\Omega_{X\/k, Q}$ implies that\n\\[\n\\sum_{g \\in G} \\res_Q((\\omega - d \\nu_Q)_g) \\cdot g \\in I_{G, G_Q}\n\\]\nby Lemma~\\ref{lem:main_lemma_Omega_Y}.\nMoreover, by~Lemma~\\ref{lem:xiQ_existence}~(3):\n\\[\n\t\\sum_{g \\in G} \\langle g^*(\\xi_Q), \\nu \\rangle \\cdot g \\in I_{G, G_Q}.\n\\]\nNote that, using Lemma~\\ref{lem:xiQ_existence}~(4), \\eqref{eqn:residue_and_trace} and the equality $\\res_P(f \\cdot dg) = - \\res_P(g \\cdot df)$:\n\\begin{align*}\n\t\\sum_{Q \\in B} \\langle g^*(\\xi_Q), \\nu \\rangle &=\n\t\\langle dg^*(z^{\\vee}), \\nu \\rangle\n\t= \\sum_{P \\in R} \\res_P(\\nu_{\\pi(P)} \\cdot dg^*(z^{\\vee}))\\\\\n\t&= - \\sum_{P \\in R} \\res_P(g^*(z^{\\vee}) \\cdot d\\nu_{\\pi(P)})\n\t= - \\sum_{Q \\in B} \\res_Q((d\\nu_Q)_g).\n\\end{align*}\nHence by the residue theorem:\n\\begin{align*}\n\t\\sum_{Q, g} (\\res_Q((\\omega - d \\nu_Q)_g) - \\langle g^*(\\xi_Q), \\nu \\rangle) \\cdot g &= \n\t\\sum_{Q, g} \\res_Q(\\omega_g) \\cdot g = 0.\t\n\\end{align*}\nWe check now that this map vanishes on $B^1_U(\\pi_* \\Omega_{X\/k}^{\\bullet})$. Suppose that $h \\in \\mc O_X(V)$. Then $(dh, (h)_Q)$\nmaps to:\n\\[\n\t\\sum_{Q, g} \\left(\\res_Q((dh - dh)_g) + \\langle g^*(\\xi_Q), (h)_Q \\rangle \\right) \\cdot g_Q = 0.\n\\]\nMoreover, if $\\nu \\in \\bigoplus_{Q \\in B} \\mc O_{X, Q}$ then by \\eqref{eqn:residue_and_trace}, \\eqref{eqn:gth_component_trace_f}\nand by Lemma~\\ref{lem:xiQ_existence} (1), (2):\n\\begin{align*}\n\t\\langle g^*(\\xi_Q), \\nu \\rangle &= \\sum_{Q' \\in B} \\sum_{P \\in \\pi^{-1}(Q')} \\res_P(\\nu_{Q'} \\cdot g^*(\\xi_Q))\\\\\n\t&= \\sum_{P \\in \\pi^{-1}(Q)} \\res_P(\\nu_Q \\cdot dg^*(z^{\\vee}))\\\\\n\t&= -\\sum_{P \\in \\pi^{-1}(Q)} \\res_P(g^*(z^{\\vee}) \\cdot d\\nu_Q)\\\\\n\t&= -\\res_Q(\\tr_{X\/Y}(g^*(z^{\\vee}) \\cdot d\\nu_Q))\\\\\n\t&= -\\res_Q((d\\nu_Q)_g).\n\\end{align*}\nHence $(0, \\nu)$ maps to:\n\\begin{equation*}\n\t\\sum_{Q, g} \\left(\\res_Q((0 - d \\nu_Q)_g) - \\langle g^*(\\xi_Q), \\nu \\rangle \\right) \\cdot g_Q\n\t= 0. \\qedhere\n\\end{equation*}\n\\end{proof}\n\\subsection*{Step IV}\n\n\tConsider the map:\n\n\t\\begin{align*}\n\t\t\\Phi_2 : M_1 &\\to M_2,\\\\\n\t\t\\Phi_2((\\omega, \\nu)) &:= \n\t\t(\\omega, \\nu)^{\\circ} + \\sum_{Q, g} \\left(\\res_Q((\\omega - d \\nu_Q)_g) - \\langle g^*(\\xi_Q), \\nu \\rangle \\right) \\cdot g_Q \\\\\n\t\t&+ e_G(\\nu), \\\\\n\t\t\\Phi_2 \\left(\\sum_{Q, g} a_{Q, g} g_Q \\right) &:= \\left(\\sum_{Q, g} a_{Q, g} g^*(z) \\eta_Q, 0 \\right),\\\\\n\t\t\\Phi_2 (\\ol{g}_Q ) &:= g^*(\\Xi_Q) - \\ol g_Q.\n\t\\end{align*}\n\n\tBy Steps I--III and Step II of the proof of Proposition~\\ref{prop:regular_and_vanishing_functions}, $\\Phi_2 : M_1 \\to M_2$ is a well-defined $k[G]$-linear map. Moreover, the Hodge--de Rham exact sequences~\\eqref{eqn:intro_hodge_de_rham_se} and~\\eqref{eqn:hdr_exact_sequence_for_log_de_rham} yield the following diagram:\n\n\t\\begin{center}\n\t\n\t\t\\begin{tikzcd}\n\t\t\t0 \\arrow[r] & {H^0(\\Omega_{X\/k}) \\oplus k[G]_B} \\arrow[r] \\arrow[d, \"\\Phi_0\"] & M_1 \\arrow[r] \\arrow[d, \"\\Phi_2\"] & {H^1(\\mc O_X) \\oplus k[G]_B^{\\vee}} \\arrow[r] \\arrow[d, \"\\Phi_1\"] & 0 \\\\\n\t\t\t0 \\arrow[r] & {H^0(\\Omega_{X\/k}(R)) \\oplus I_{X\/Y}} \\arrow[r]                 & M_2 \\arrow[r]                     & {H^1(\\mc O_X(-R)) \\oplus J_{X\/Y}} \\arrow[r]                       & 0\n\t\t\\end{tikzcd}\n\t\\end{center}\n\n\tThe proof follows by noting that $\\Phi_0$ and $\\Phi_1$ are isomorphisms by Propositions~\\ref{prop:log_diffs_and_diffs}\n\tand~\\ref{prop:regular_and_vanishing_functions}.\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main_thm}, part 2]\n\n\tLet $K_1$ be the kernel of the epimorphism:\n\n\t\\begin{alignat*}{3}\n\t\tH^1_{dR, R}(X\/k) &\\to \\quad H^1(X, \\mc O_X(-R)) &&\\to H^1 \\left(Y, \\bigoplus_{g \\in G} g^*(z^{\\vee}) \\mc O_Y(-B) \\right)&\\\\\n\t\t&\\cong \\Ind^G H^1(Y, \\mc O_Y(-B)) &&\\cong k[G]^{g_Y + \\# B - 1},&\n\t\\end{alignat*}\n\n\twhere the first map comes from the Hodge--de Rham exact sequence~\\eqref{eqn:hdr_exact_sequence_for_log_de_rham}\n\tand the second map comes from the inclusion from the Lemma~\\ref{lem:inclusions_of_modules2}\n\t(note that the second map is surjective, since the first cohomology of a finitely supported sheaf on a curve\n\tvanishes).\n\tThen, since $k[G]$ is an injective $k[G]$-module (cf.~\\cite[Corollary 8.5.3]{Webb_finite_group_representations}):\n\n\t\\begin{equation} \\label{eqn:H1_dR=K1+k[G]}\n\t\tH^1_{dR, R}(X\/k) \\cong K_1 \\oplus k[G]^{g_Y + \\# B - 1}.\n\t\\end{equation}\n\n\tMoreover, one can explicitly describe $K_1$ as:\n\n\t\\[\n\t\tK_1 = \\frac{\\{ (\\omega, \\nu) : \\omega - d\\nu_Q \\in \\Omega_{X\/k}(Q)_Q \\quad \\forall_{Q \\in B} \\}}{\\{ (df, (f + f_Q))_Q \\}},\n\t\\]\n\n\twhere:\n\n\t\\begin{itemize}\n\t\t\\item $\\omega \\in \\Omega_{X\/k}(V)$,\n\t\t\\item $\\nu \\in \\bigoplus_{Q \\in B} \\bigoplus_{g \\in G} g^*(z) \\mc O_Y(-B)_Q$,\n\t\t\\item $f \\in \\mc O_X(V) \\cap \\bigcap_{Q \\in B} \\bigoplus_{g \\in G} g^*(z) \\mc O_Y(-B)_Q$,\n\t\t\\item $f_Q \\in \\mc O_X(-R)_Q$.\n\t\\end{itemize}\n\n\tHowever,\n\n\t\\begin{align*}\n\t\t\\mc O_X(V) \\cap \\bigcap_{Q \\in B} \\bigoplus_{g \\in G} g^*(z) \\mc O_Y(-B)_Q &= \\bigoplus_{g \\in G} g^*(z) \\mc O_Y(U) \\cap \n\t\t\\bigcap_{Q \\in B} \\bigoplus_{g \\in G} g^*(z) \\mc O_Y(-B)_Q\\\\\n\t\t&= \\bigoplus_{g \\in G} g^*(z) \\mc O_Y(-B)(Y) = 0.\n\t\\end{align*}\n\n\tHence:\n\n\t\\begin{align*}\n\t\tK_1 &= \\frac{\\{ (\\omega, \\nu) : \\omega - d\\nu_Q \\in \\Omega_{X\/k}(Q)_Q \\quad \\forall_{Q \\in B} \\}}{\\{ (0, (f_Q)_Q) : f_Q \\in \\mc O_X(-R)_Q\n\t\t\t\\quad \\forall_{Q \\in B} \\}}\\\\\n\t\t&= \\{ (\\omega \\in \\Omega_{X\/k}(V), \\nu \\in \\bigoplus_{Q \\in B} H^1_Q) : \\omega - d\\nu_Q \\in \\Omega_{X\/k}(Q)_Q \\quad \\forall_{Q \\in B} \\}.\n\t\\end{align*}\t\n\n\tThe Hodge--de Rham exact sequence~\\eqref{eqn:hdr_exact_sequence_for_log_de_rham} implies that the composition\n\n\t\\[\n\t\tH^0 \\left(Y, \\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B) \\right) \\to H^0(X, \\Omega_{X\/k}(R)) \\to H^1_{dR, R}(X\/k)\n\t\\]\n\n\tfactors through the monomorphism:\n\n\t\\begin{equation} \\label{eqn:monomorphism_K1}\n\t\tH^0 \\left(Y, \\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B) \\right) \\to K_1.\n\t\\end{equation}\n\n\tLet $K_2$ be the cokernel of the monomorphism~\\eqref{eqn:monomorphism_K1}. Then, analogously as before:\n\n\t\\begin{align}\n\t\tK_1 &\\cong H^0 \\left(Y, \\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B) \\right) \\oplus K_2 \\label{eqn:K1=K2+k[G]}\\\\\n\t\t&\\cong k[G]^{g_Y + \\# B - 1} \\oplus K_2. \\nonumber\n\t\\end{align}\n\n\tOn the other hand:\n\n\t\\begin{align*}\n\t\tK_2 &= \\frac{\\{ (\\omega \\in \\Omega_{X\/k}(V), \\nu) : \\omega - d\\nu_Q \\in \\Omega_{X\/k}(Q)_Q, \\quad \\nu_Q \\in H^1_Q\n\t\t\t\\quad \\forall_{Q \\in B} \\}}{\\{ (\\omega, 0) : \\omega \\in \\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B)(Y) \\}}\\\\\n\t\t&\\cong \\{ (\\omega, \\nu) : \\omega \\in \n\t\t\t\\frac{\\Omega_{X\/k}(V)}{\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B)(Y)}, \\quad\n\t\t\t\\omega - d\\nu_Q \\in \\Omega_{X\/k}(Q)_Q, \\quad \\nu_Q \\in H^1_Q \\quad \\forall_{Q \\in B} \\}.\n\t\\end{align*}\n\n\tLet $j : V \\to X$ be the open immersion.\n\tNote that both sheaves:\n\n\t\\[\n\t\t\\frac{\\pi_* \\Omega_{X\/k}(R)}{\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B)} \\textrm{ and } \n\t\t\\frac{(\\pi \\circ j)_* \\, \\Omega_{V\/k}}{\\pi_* \\Omega_{X\/k}(R)}\n\t\\]\n\n\thave support contained in $B$. Hence $(\\pi \\circ j)_* \\Omega_{V\/k}\/\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B)$ is also supported on $B$.\n\tThis implies that:\n\n\t\\[\n\t\t\\frac{\\Omega_{X\/k}(V)}{\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B)(Y)} \\cong \\bigoplus_{Q \\in B}\n\t\t\\frac{\\Omega_{k(X)\/k}}{\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B)_Q}.\n\t\\]\n\n\tTherefore:\n\n\t\\[\n\t\tK_2 \\cong \\bigoplus_{Q \\in B} H^1_{dR, Q},\n\t\\]\n\n\twhere:\n\n\t\\begin{align}\n\t\tH^1_{dR, Q} &:= \\{ (\\omega_Q \\in \\frac{\\Omega_{k(X)\/k}}{\\bigoplus_{g \\in G} g^*(z) \\Omega_{Y\/k}(B)_Q}, \\nu_Q \\in H^1_Q) :\n\t\t\\quad \\omega_Q - d\\nu_Q \\in \\Omega_{X\/k}(R)_Q \\}. \\label{eqn:H1dRQ}\n\t\\end{align}\n\n\tThus by~\\eqref{eqn:H1_dR=K1+k[G]} and~\\eqref{eqn:K1=K2+k[G]}:\n\n\t\\begin{align*}\n\t\tH^1_{dR, R}(X\/k) &\\cong K_1 \\oplus k[G]^{g_Y + \\# B - 1}\\\\\n\t\t&\\cong K_2 \\oplus k[G]^{2 \\cdot (g_Y + \\# B - 1)}\\\\\n\t\t&\\cong \\bigoplus_{Q \\in B} H^1_{dR, Q} \\oplus k[G]^{2 \\cdot (g_Y + \\# B - 1)}.\n\t\\end{align*}\n\n\tOne finishes the proof using Proposition~\\ref{prop:normal_and_log_de_rham}, similarly as in Section~\\ref{sec:OmegaX}.\n\n\\end{proof}\nIn the sequel we will need to know the relation between $H^0_Q$, $H^1_Q$, $H^1_{dR, Q}$. This is provided\nby the following local analogue of the Hodge--de Rham exact sequence.\n\\begin{Lemma} \\label{lem:properties_IXY_H1dRQ}\n\n\tFor every $Q \\in B$ there exists an exact sequence:\n\n\t\\begin{equation*}\n\t\t0 \\to H^0_Q \\to H^1_{dR, Q} \\to H^1_Q \\to 0.\n\t\\end{equation*}\n\\end{Lemma}\n\\begin{proof}\n\tIt is straightforward that the map $H^0_Q \\to H^1_{dR, Q}$ induced\n\tby $\\omega_Q \\mapsto (\\omega_Q, 0)$ is injective. The map $H^1_{dR, Q} \\to H^1_Q$, $(\\omega_Q, \\nu_Q) \\mapsto \\nu_Q$ is surjective, since $\\nu_Q$ is the image of $(d \\nu_Q, \\nu_Q)$. Finally, $(\\omega_Q, \\nu_Q)$ is\n\tin the kernel of the map $H^1_{dR, Q} \\to H^1_Q$ if and only if\n\t$\\nu_Q \\in \\mc O_X(-R)_Q$, which is equivalent to $\\omega_Q \\in \\Omega_{X\/k}(R)$ (since $\\omega_Q - d \\nu_Q \\in \\Omega_{X\/k}(R)_Q$).\n\n\\end{proof}\nWe prove now the corollary concerning the structure of $H^1_{dR}(X\/k)$ in the weak ramification case.\n\\begin{proof}[Proof of Corollary~\\ref{cor:hdr_exact_sequence}]\n\tThe proof of~\\cite[Main Theorem]{Garnek_equivariant} shows that if $H^1_{dR}(X\/k) \\cong H^0(X, \\Omega_{X\/k}) \\oplus H^1(X, \\mc O_X)$ as $k[G]$-modules, then $d_P'' = 0$ for every $P \\in X(k)$.\n\tSuppose now that $d_P'' = 0$ for every $P \\in X(k)$. Then\n\t$H^0_Q = H^1_Q = H^1_{dR, Q} = 0$ for every $Q \\in Y(k)$ by Lemma~\\ref{lem:properties_H0Q_H1Q} and Lemma~\\ref{lem:properties_IXY_H1dRQ}.\n\tHence:\n\n\t\\[\n\t\tH^1_{dR}(X\/k) \\cong k[G]^{\\oplus 2 \\cdot g_Y} \\oplus I_{X\/Y} \\oplus J_{X\/Y}\n\t\t\\cong H^0(X, \\Omega_{X\/k}) \\oplus H^1(X, \\mc O_X). \\qedhere\n\t\\]\n\n\\end{proof}\n\n\\section{Artin--Schreier covers} \\label{sec:AS_covers}\nIn this subsection we construct a magical element for a large class of Artin--Schreier covers\nand compute their cohomology. Also, we give an example of a $\\ZZ\/p$-cover without a magical element.\\\\\n\nLet $k$ be an algebraically closed field of characteristic $p$ and $Y\/k$ be a smooth projective curve.\nRecall that the $\\ZZ\/p$-covers\nof $Y$ are in a bijection with the group $k(Y)\/\\wp(k(Y))$, where $\\wp(f) := f^p - f$. An element of\n$k(Y)\/\\wp(k(Y))$ represented by a function $f \\in k(Y)$ corresponds to a curve\nwith the function field given by the equation:\n\\begin{equation} \\label{eqn:artin_schreier}\n\ty^p - y = f.\n\\end{equation}\nThe action of $G = \\langle \\sigma \\rangle \\cong \\ZZ\/p$ on $X$ is \nthen given by $\\sigma(y) := y+1$. We say that $y$ is an \\bb{Artin--Schreier generator} of $\\pi$.\nFrom now on we assume that $\\pi : X \\to Y$ is a $\\ZZ\/p$-cover given by an equation\nof the form~\\eqref{eqn:artin_schreier}.\n\\subsection{Local standard form}\nKeep the above setup. An Artin--Schreier generator $y$ is in \\bb{local standard form at $Q \\in Y(k)$},\nif it satisfies an equation of the form~\\eqref{eqn:artin_schreier}, in which either $f$ is regular at $Q$ or $f$ has a pole of order not divisible by $p$ at $Q$.\\\\\n\nNote that for every $Q \\in Y(k)$, there exists an Artin--Schreier generator of $X$ in local standard form at~$Q$. Indeed, given an arbitrary equation of the form~\\eqref{eqn:artin_schreier} one can\nrepeatedly replace $y$ by $y - g$ and $f$ by $f - \\wp(g)$, where $g$ is a power\nof a uniformizer at~$Q$.\nSuppose now that $y$ is in local standard form at $Q$. In this situation we denote:\n\\[\nm_Q = m_{X\/Y, Q} := \n\\begin{cases}\n\t|\\ord_Q(f)|, & \\textrm{ if } \\ord_Q(f) < 0,\\\\\n\t0, & \\textrm{ otherwise.}\n\\end{cases}\n\\]\n(one checks that $m_Q$ does not depend\non the choice of $y$). It turns out that:\n\\[\nm_Q = \\max \\{ i : G_{Q, i} \\neq 0 \\}\n\\]\nand hence $d_Q = (m_Q + 1) \\cdot (p-1)$ (cf.~\\cite[Lemma 4.2]{Garnek_equivariant}).\nThe following result allows us to compute $m_Q$ without finding local standard form in most cases.\n\\begin{Lemma} \\label{lem:computing_mQ}\n\n\tLet $\\pi : X \\to Y$ be a $\\ZZ\/p$-cover given by~\\eqref{eqn:artin_schreier}.\n\tSuppose that $Q \\in Y(k)$ is a pole of $f$ satisfying:\n\n\t\\begin{equation} \\label{eqn:assumption_to_compute_mQ}\n\t\t\\ord_Q(df) < \\frac{1}{p} \\ord_Q(f) - 1.\n\t\\end{equation}\n\n\tThen $m_Q = -\\ord_Q(df) - 1$.\n\n\\end{Lemma}\n\\begin{proof}\n\n\tNote that $f = \\wp(g) + h$, where $\\ord_Q(g) = \\ord_Q(f)\/p$ and:\n\n\t\\[ \\ord_Q(h) = \n\t\\begin{cases}\n\t\t-m_Q, & \\textrm{ if } m_Q \\ge 1,\\\\\n\t\t\\ge 0, & \\textrm{ if } m_Q = 0.\n\t\\end{cases}\n\t\\]\n\n\tThen $df + dg = dh$. Observe that $\\ord_Q(dg) \\ge \\ord_Q(g) - 1 = \\frac{1}{p} \\ord_Q(f) - 1 > \\ord_Q(df)$\n\tand hence:\n\n\t\\[\n\t-m_Q - 1 = \\ord_Q(dh) = \\min(\\ord_Q(df), \\ord_Q(dg)) =  \\ord_Q(df). \\qedhere\n\t\\]\n\n\\end{proof}\n\n\\subsection{Global standard form} \\label{subsec:gsf}\nAn Artin--Schreier generator $y$ is said to be in \\bb{global standard form}, if it\nis in local standard form at every $Q \\in Y(k)$ and $y \\not \\in k$. In this situation we say also that $\\pi$ has a global standard form. The following result explains our interest in this notion.\n\\begin{Lemma} \\label{lem:gsf_gives_me_for_Zp}\n\tSuppose that $y$ is an Artin--Schreier generator in global standard form for~$\\pi$.\n\tThen $z := y^{p-1}$ is a magical element for\n\t$\\pi$ and the dual of $z$ with respect to the trace pairing is $z^{\\vee} := 2 - z$.\n\\end{Lemma}\n\\begin{proof}\n\tNote that $\\ord_P(z) = (p-1) \\cdot \\ord_{P}(y) = - (p-1) \\cdot m_P = -d_P$\n\tfor every $P \\in R$. Moreover, one checks that\n\n\t\\begin{equation} \\label{eqn:AS_trace_of_yi}\n\t\t\t\\tr_{X\/Y}(y^i) =\n\t\t\\begin{cases}\n\t\t\t0, & \\textrm{ for } 0 \\le i < p-1 \\textrm{ and } p-1 < i < 2 \\cdot (p-1),\\\\\n\t\t\t-1, & \\textrm{ for } i = p - 1 \\textrm{ and } i = 2 \\cdot (p-1).\n\t\t\\end{cases}\n\t\\end{equation}\n\n\tThis allows us to conclude that $\\tr_{X\/Y}(z) \\neq 0$ and $z^{\\vee}$ is the dual element to~$z$.\n\\end{proof}\n\nNot every $\\ZZ\/p$-cover has a global standard form. For example,\nconnected \\'{e}tale $\\ZZ\/p$-covers do not have a global standard form (cf.~\\cite[Subsection~3.1]{WardMarques_HoloDiffs}). We present another example in Subsection~\\ref{subsec:no_magical_element}.\nIt turns out that every sufficiently ramified $\\ZZ\/p$-cover has a global standard form.\n\\begin{Lemma} \\label{lem:criterion_for_gsf}\n\n\tSuppose that there exists a point $Q_0 \\in Y(k)$ with $m_{Q_0} > 2g_Y \\cdot p$. Then the cover $\\pi$\n\thas a global standard form.\n\\end{Lemma}\n\\begin{proof}\n\n\tLet $y$ be an Artin--Schreier generator for $\\pi$ in local standard form at $Q_0$ and let $y^p - y = f$.\n\tSuppose that $\\ord_Q(f) = -j < 0$, $p|j$ for some $Q \\in Y(k)$. Using the Riemann--Roch theorem, we may choose a function\n\t$g \\in k(Y)$ such that:\n\n\t\\begin{itemize}\n\t\t\\item $\\ord_{Q_0}(g) = -2g_Y$,\n\t\t\\item $\\ord_Q(g) = -j\/p$,\n\t\t\\item $\\ord_{Q'}(g) \\ge 0$ for $Q' \\neq Q, Q_0$.\n\t\\end{itemize}\n\n\tNote that there exists $c \\in k$ such that $\\ord_Q(f - \\wp(c \\cdot g)) > -j$. Let $y_1 := y - c \\cdot g$, $f_1 := f - \\wp(c \\cdot g)$. Then $y_1$ is an Artin--Schreier generator\n\tof $\\pi$. Moreover:\n\n\t\\begin{itemize}\n\t\t\\item $y_1$ is in local standard form at $Q_0$ -- indeed, $\\ord_{Q_0}(f_1) = \\ord_{Q_0}(f)$, since $\\ord_{Q_0}(\\wp(g)) = -2g_Y \\cdot p > - m_{Q_0}$,\n\t\t\n\t\t\\item if $y$ is in local standard form at $Q' \\neq Q, Q_0$,\n\t\tthen $y_1$ is also in local standard form at $Q'$,\n\t\t\n\t\t\\item $\\ord_Q(f_1) > \\ord_Q(f)$.\n\t\\end{itemize}\n\n\tThus, by repeatedly replacing $(y, f)$ by $(y_1, f_1)$, we eventually obtain an Artin--Schreier generator of $\\pi$ in global standard form.\n\\end{proof}\n\\subsection{Proof of Corollary~\\ref{cor:cohomology_of_Zp}} \\label{sec:Zp}\nIn this subsection we apply Theorem~\\ref{thm:main_thm} to prove Corollary~\\ref{cor:cohomology_of_Zp}.\nKeep the notation of Corollary~\\ref{cor:cohomology_of_Zp}. Additionally, we assume that\n$X$ satisfies an equation of the form~\\eqref{eqn:artin_schreier}, where $y$ is in global standard form.\nBy Lemma~\\ref{lem:gsf_gives_me_for_Zp} $z := y^{p-1}$ is a magical element for $\\pi$ and we may take $z^{\\vee} = 2 - z$.\nTherefore:\n\\[\n\\Span_k(g^*(z) : g \\in G) = \\Span_k(g^*(z^{\\vee}) : g \\in G) = \\Span_k(1, y, \\ldots, y^{p-1}).\n\\]\nRecall that every finitely dimensional indecomposable $k[G]$-module is of the form\n$J_i = k[x]\/(x - 1)^i$ for some $i = 1, \\ldots, p$, where $G$ acts by multiplication by $x$ (cf. \\cite[Theorem 12.1.5]{DummitFoote2004}). Note that as a $k[G]$-module\n$\\Span_k(1, y, \\ldots, y^{i-1}) \\cong J_i$ for any $1 \\le i \\le p$.\\\\\n\n\nFix $Q \\in B$ and denote $m := m_Q$, $\\pi^{-1}(Q) = \\{ P \\}$.\nIn order to prove Corollary~\\ref{cor:cohomology_of_Zp}, it suffices to compute bases of the modules $H^0_Q$, $H^1_Q$, $H^1_{dR, Q}$, defined by~\\eqref{eqn:H0Q}, \\eqref{eqn:H1Q} and~\\eqref{eqn:H1dRQ} respectively.\nTo this end, we need to pick an appropriate uniformizer at $Q$. By approximating\nthe element $1\/\\sqrt[m]{f} \\in \\wh{\\mc O}_{Y, Q}$, we can choose $t \\in k(Y)$ such that:\n\\[\n\\frac{1}{t^m} \\equiv y^p - y \\pmod{\\mf m_{Y, Q}^{2pm}}.\n\\]\n\\begin{Proposition} \\label{prop:H0Q_for_Zp}\n\tKeep the above setup.\n\t\\begin{enumerate}[(1)]\n\t\t\\item The images of the elements:\n\t\n\t\t\\[\n\t\ty^i \\frac{dt}{t^j} \\in \\Omega_{k(X)\/k},\n\t\t\\]\n\t\n\t\twhere $0 \\le i \\le p-1$, $2 \\le j$, $mi + pj \\le (m+1) \\cdot (p-1) + 1$, form a basis of $H^0_Q$.\n\t\n\t\t\\item The images of the elements:\n\t\n\t\t\\[\n\t\ty^{i} \\cdot t^{j} \\in k(X), \n\t\t\\]\n\t\n\t\twhere $0 \\le i \\le p-1$, $1 \\le j$, $-mi + pj < 1$ form a basis of $H^1_Q$.\n\t\n\t\t\\item The images of the elements:\n\t\n\t\t\\[\n\t\t\\left(\\frac{y^i \\, dt}{t^j}, \\frac{1}{(i+1) \\cdot m} y^{i+1} t^{m+1 - j} \\right) \\in \\Omega_{k(X)\/k} \\times k(X),\n\t\t\\]\n\t\n\t\twhere $0 \\le i \\le p-2$, $2 \\le j \\le m$, form a basis of $H^1_{dR, Q}$.\n\t\\end{enumerate}\n\\end{Proposition}\nBefore the proof, note that for any $\\omega \\in \\Omega_{k(X)\/k}$ there exists a\nunique system of forms $\\omega_0, \\ldots, \\omega_{p-1} \\in \\Omega_{k(Y)\/k}$ such that\n$\\omega = \\omega_0 + y \\cdot \\omega_1 + \\ldots + y^{p-1} \\cdot \\omega_{p-1}$.\nMoreover, one has:\n\\begin{equation} \\label{eqn:valuation_of_omega_Zp}\n\t\\ord_P(\\omega) = \\min \\{ \\ord_P(y^i \\omega_i) : i = 0, \\ldots, p-1 \\}.\n\\end{equation}\nIndeed, for any $0 \\le i < j \\le p-1$:\n\\begin{align*}\n\t\\ord_P(y^i \\cdot \\omega_i) - \\ord_P(y^j \\cdot \\omega_j) &= (-i m + p \\cdot \\ord_Q(\\omega_i) + d_Q)\\\\\n\t&- (-j m + p \\cdot \\ord_Q(\\omega_j) + d_Q)\\\\\n\t&= m \\cdot(j - i) + p \\cdot (\\ord_Q(\\omega_i) - \\ord_Q(\\omega_j)) \\neq 0,\n\\end{align*}\nsince $p \\nmid m \\cdot (i - j)$. Analogously, any $f \\in k(X)$ is of the form\n$f = f_0 + y \\cdot f_1 + \\ldots + y^{p-1} \\cdot f_{p-1}$\nfor some $f_0, \\ldots, f_{p-1} \\in k(Y)$ and\n\\begin{equation} \\label{eqn:valuation_of_f_Zp}\n\t\\ord_P(f) = \\min \\{ \\ord_P(y^i f_i) : i = 0, \\ldots, p-1 \\}.\n\\end{equation}\n\\begin{proof}[Proof of Proposition~\\ref{prop:H0Q_for_Zp}]\n\t(1) Let for any $i$, $j_{max}(i)$ denote the largest integer satisfying\n\t\t$mi + pj \\le (m+1) \\cdot (p-1) + 1$. By~\\eqref{eqn:valuation_of_omega_Zp} and~\\eqref{eqn:valuation_of_diff_form}:\n\t\n\t\t\\begin{align*}\n\t\t\t\\omega \\in \\Omega_{X\/k}(R)_Q &\\Leftrightarrow\n\t\t\t\\ord_P(y^i \\cdot \\omega_i) \\ge -1 \\textrm{ for every } i = 0, \\ldots, p-1\\\\\n\t\t\t&\\Leftrightarrow \\ord_Q(\\omega_i) \\ge -j_{max}(i).\n\t\t\\end{align*}\n\t\n\t\tHence:\n\t\n\t\t\\begin{equation*}\n\t\t\tH^0_Q \\cong \\bigoplus_{i = 0}^{p-1} \n\t\t\ty^i \\cdot \\frac{t^{-j_{max}(i)} \\Omega_{Y\/k, Q}}{\\Omega_{Y\/k, Q}}.\n\t\t\\end{equation*}\n\t\n\t\tThe statement follows easily by noting that the images of the forms $dt\/t^j$\n\t\tfor $j = 2, \\ldots, j_{max}(i)$ form a basis of $\\frac{t^{-j_{max}(i)} \\Omega_{Y\/k, Q}}{\\Omega_{Y\/k, Q}}$.\n\t\t\n\t(2) Let $j_{min}(i)$ denote the least integer satisfying the inequality $pj - mi \\ge 1$. Analogously, using \\eqref{eqn:valuation_of_f_Zp} one obtains:\n\t\n\t\t\\begin{equation*}\n\t\t\tH^1_Q \\cong \\bigoplus_{i = 0}^{p-1} y^i \\frac{\\mc O_Y(-R)_Q}{t^{j_{min}(i)} \\mc O_Y(-R)_Q}.\n\t\t\\end{equation*}\n\t\n\t\tTo finish the proof of (2), it suffices to notice that the images of the elements\n\t\t$t^j$ for $j = 1, \\ldots, j_{min}(i)-1$ form a basis of $\\frac{\\mc O_Y(-R)_Q}{t^{j_{min}(i)} \\mc O_Y(-R)_Q}$.\n\t\t\n\t(3) By the choice of $t$ we have $dy \\equiv m \\cdot dt\/t^{m+1} \\pmod{\\mf m_{Y, Q}^{2pm - 1} \\Omega_{Y, Q}}$,\n\t\twhich easily implies that:\n\t\n\t\t\\[\n\t\ty^i \\cdot t^{m+1 - j} \\, dy \\equiv m \\cdot y^i dt\/t^j \\pmod{\\Omega_{Y\/k, Q}(R)}\t\n\t\t\\]\n\t\n\t\tThus:\n\t\n\t\t\\begin{align*}\n\t\t\t\\frac{y^i \\, dt}{t^j} - d \\left( \\frac{1}{(i+1) \\cdot m} y^{i+1} t^{m+1 - j} \\right)\n\t\t\t&\\equiv - \\frac{(m+1 - j)}{(i+1) \\cdot m} y^{i+1} t^{m - j} \\, dt\\\\\n\t\t\t&\\equiv 0 \\pmod{\\Omega_{Y\/k, Q}(R)},\n\t\t\\end{align*}\n\t\n\t\tsince:\n\t\n\t\t\\begin{align*}\n\t\t\t\\ord_P(y^{i+1} t^{m - j} \\, dt) &= -(i+1) \\cdot m + (m-j) \\cdot p\n\t\t\t+ (p-1) \\cdot (m+1)\\\\\n\t\t\t&\\ge -(p-1) \\cdot m + 0 \\cdot p\n\t\t\t+ (p-1) \\cdot (m+1)\\\\\n\t\t\t&= p-1 \\ge 0.\n\t\t\\end{align*}\n\t\n\t\tHence the listed elements define valid elements of $H^1_{dR, Q}$. Moreover:\n\t\n\t\t\\begin{itemize}\n\t\t\t\\item part (1) implies that the elements\n\t\t\n\t\t\t\\[\n\t\t\t\\left(\\frac{y^i \\, dt}{t^j}, \\frac{1}{(i+1) \\cdot m} y^{i+1} t^{m+1 - j} \\right) =\n\t\t\t\\left(\\frac{y^i \\, dt}{t^j}, 0 \\right) \\qquad \\textrm{ in } H^1_{dR, Q}\n\t\t\t\\]\n\t\t\n\t\t\tfor $i = 0, \\ldots, p-1$ and $j = 2, \\ldots, j_{max}(i)$,\n\t\t\tare images of a basis of $H^0_Q$ through the map $H^0_Q \\to H^1_{dR, Q}$ from Lemma~\\ref{lem:properties_IXY_H1dRQ},\n\t\t\t\n\t\t\t\\item part (2) implies that the images of the elements:\n\t\t\n\t\t\t\\[\n\t\t\t\\left(\\frac{y^i \\, dt}{t^j}, \\frac{1}{(i+1) \\cdot m} y^{i+1} t^{m+1 - j} \\right)\n\t\t\t\\]\n\t\t\n\t\t\tfor $i = 0, \\ldots, p-1$ and $j = j_{max}(i) + 1, \\ldots, m$ through the map $H^1_{dR, Q} \\to H^1_Q$ form a basis\n\t\t\tof $H^1_Q$ (one easily checks that $j_{min}(i+1) = m+1 - j_{max}(i)$).\n\t\t\\end{itemize}\n\t\n\t\tTherefore the listed elements form a basis of $H^1_{dR, Q}$ by Lemma~\\ref{lem:properties_IXY_H1dRQ}~(2).\n\\end{proof}\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:cohomology_of_Zp}]\n\n\tLet for any $j$, $i_{max}(j)$ denote the largest integer satisfying\n\t$mi + pj \\le (m+1) \\cdot (p-1) + 1$. By Proposition~\\ref{prop:H0Q_for_Zp}:\n\n\t\\begin{align*}\n\t\tH^0_Q = \\bigoplus_{j \\ge 2} \\frac{dt}{t^j} \\cdot \\Span_k(1, y, y^2, \\ldots, y^{i_{max}(j)})\n\t\t\\cong \\bigoplus_{j \\ge 2} J_{i_{max}(j) + 1}.\n\t\\end{align*}\n\n\tIt is an elementary calculation to check that for $0 \\le i \\le p-1$:\n\n\t\\begin{align*}\n\t\t\\# \\{j \\ge 2 : i_{max}(j) + 1 = i \\} =\\alpha(i).\n\t\\end{align*}\n\n\tA similar computation allows us to find $H^1_{dR, Q}$. The proof follows by Theorem~\\ref{thm:main_thm}.\n\\end{proof}\n\\subsection{$\\ZZ\/p$-cover without a magical element} \\label{subsec:no_magical_element}\nThe goal of this subsection is to construct a $\\ZZ\/p$-cover without a magical element\n(and thus without a global standard form).\nLet $p \\ge 5$ be a prime and $k = \\overline{\\FF}_p$. Pick a general smooth projective curve $Y\/k$ of genus $g_Y \\ge p\/2$.\nThen its gonality equals $\\gamma = \\lfloor \\frac{g_Y + 3}{2} \\rfloor$ by Brill--Noether theory, see e.g.~\\cite[Proposition 1.1. (v)]{Poonen_Gonality_Modular}. \nLet $m$ and $M$ be integers satisfying $p \\nmid m$ and:\n\\[\n\t\\frac{2 g_Y}{p} < \\frac{m}{p} < M < \\gamma.\n\\]\nFix a point $Q \\in Y(k)$ and let $t \\in k(Y)$ be a uniformizer at $Q$. Using Riemann--Roch theorem,\nwe may choose a function $f \\in k(Y)$ that is regular outside of $Q$ and satisfies:\n\\[\n\tf \\equiv \\frac{1}{t^{Mp}} - \\frac{1}{t^M} + \\frac{1}{t^m} \\pmod{\\mf m_{Y, Q}^{-2g_Y}}.\n\\]\nLet $\\pi : X \\to Y$ be the $\\ZZ\/p$-cover defined by the equation~\\eqref{eqn:artin_schreier}. \nSuppose to the contrary that $z = \\sum_{i = 0}^{p-1} z_i \\cdot y^i \\in k(X)$\nis a magical element. By~\\eqref{eqn:trace_of_dual_z} we may assume that $z_{p-1} = -\\tr_{X\/Y}(z) = 1$. Let $\\pi^{-1}(Q) = \\{ P \\}$. Note that for $Q' \\in Y(k)$, $Q' \\neq Q$, we have $\\ord_{Q'}(y) \\ge 0$.\nTherefore, by~\\eqref{eqn:AS_trace_of_yi} and~\\eqref{eqn:valuation_of_f_Zp}:\n\\[\n\tz_{p-2} = - \\tr_{X\/Y}(y \\cdot z) + \\tr_{X\/Y}(y^p) \\in \\mc O_{Y, Q'}.\n\\]\nObserve also that $y_1 := y - \\frac{1}{t^M}$ is in local standard form at $Q$ and $m_Q = m$. Since $z = y_1^{p-1} + y_1^{p-2} \\cdot (z_{p-2} - \\frac{1}{t^M}) + \\ldots$, we deduce\nusing~\\eqref{eqn:valuation_of_f_Zp}:\n\\begin{align*}\n\t\\ord_P(z) \\ge -d_P' &\\Rightarrow \\ord_P \\left(y_1^{p-2} \\cdot (z_{p-2} - \\frac{1}{t^M}) \\right) \\ge -(p-1) \\cdot m\\\\\n\t&\\Rightarrow \\ord_Q \\left(z_{p-2} - \\frac{1}{t^M} \\right) \\ge -m\/p.\n\\end{align*}\nThus $z_{p-2} \\in H^0(Y, \\mc O_Y(M \\cdot Q))$, $z_{p-2} \\not \\in k$, which yields\na contradiction with $M < \\gamma$.\n\n\\section{Constructing magical elements} \\label{sec:constructing_magical_elements}\nIn this section we show that a generic $p$-group cover has a magical element.\nWe prove this using the global standard form. The following observation shows that everything comes down to\nconstructing magical elements for $\\ZZ\/p$-covers.\n\\begin{Lemma} \\label{lem:new_magical_elts}\n\tSuppose that $G$ is a finite $p$-group and $k$ is an algebraically closed field of characteristic $p$.\n\tLet $\\pi : X \\to Y$ be a $G$-cover of smooth projective curves over~$k$. Suppose that $\\pi$ factors\n\tthrough a Galois cover $X' \\to Y$. If $z_1 \\in k(X)$ is a magical element for the cover $X \\to X'$ and $z_2 \\in k(X')$ \n\tis a magical element for $X' \\to Y$ then $z_1 \\cdot z_2$ is a magical element\n\tfor $X \\to Y$.\n\\end{Lemma}\n\\begin{proof}\n\n\tLet $P \\in \\pi^{-1}(Q)$ and let $P'$ be the image of $P$ on $X'$. Then:\n\n\t\\begin{align*}\n\t\t\\ord_P(z_1 \\cdot z_2) &= \\ord_P(z_1) + e_{X\/X', P} \\cdot \\ord_{P'}(z_2)\\\\\n\t\t&\\ge -d_{X\/X', P}' - e_{X\/X', P} \\cdot d_{X'\/Y, P'}' = -d_P'.\n\t\\end{align*}\t\n\n\tMoreover, since $\\tr_{X\/X'}(z_1), \\tr_{X'\/Y}(z_2) \\in k^{\\times}$ by~\\eqref{eqn:trace_of_dual_z}:\n\n\t\\begin{align*}\n\t\t\\tr_{X\/Y}(z_1 \\cdot z_2) = \\tr_{X'\/Y}( \\tr_{X\/X'}(z_1 \\cdot z_2))\n\t\t= \\tr_{X\/X'}(z_1) \\cdot \\tr_{X'\/Y}(z_2) \\neq 0.\n\t\\end{align*}\n\n\tThis ends the proof.\n\\end{proof}\nLet $k$, $G$, $\\pi : X \\to Y$ be as in Lemma~\\ref{lem:new_magical_elts}. We say that $\\pi$ has a \\bb{global standard form}, if there exist $y_1, \\ldots, y_n \\in k(X)$ and a composition series\n\\begin{equation*} \\label{eqn:composition_series}\n0 = G_0 \\unlhd G_1 \\unlhd \\ldots \\unlhd G_n = G\n\\end{equation*}\nsuch that $y_i$ is the Artin Schreier generator in global standard form of the $\\ZZ\/p$-cover $X\/G_{i - 1} \\to X\/G_i$\nfor $i = 1, \\ldots, n$. Lemmas~\\ref{lem:gsf_gives_me_for_Zp} and~\\ref{lem:new_magical_elts} imply that in this case $z := y_1^{p-1} \\cdot \\ldots \\cdot y_n^{p-1} \\in k(X)$ is a magical element for $\\pi$. The notion of a global standard form of a $p$-group cover appeared already in~\\cite{WardMarques_HoloDiffs}, where it was used to construct a basis of holomorphic differentials of $X$ in some cases.\n\n\\subsection{Generic $p$-group covers}\nThe goal of this subsection is to prove Theorem~\\ref{thm:generic_intro}. Before the proof we review briefly the theory of moduli of $p$-group covers from~\\cite{Harbater_moduli_of_p_covers}.\\\\\n\nLet $U$ be a non-empty affine open of a smooth\nprojective curve $Y$ over $k$ and fix $u \\in U(k)$. Denote $B := Y(k) \\setminus U(k)$. Let $M_{U, G}$ be the moduli space of\npointed \\'{e}tale $G$-covers of $(U, u)$, as defined by Harbater in~\\cite{Harbater_moduli_of_p_covers}. Note that such covers correspond bijectively to pointed covers of $(Y, u)$ unramified over $U$.\\\\\n\nIn order to describe the structure of $M_{U, G}$, one proceeds inductively.\nLet $H \\cong \\ZZ\/p$ be a central normal subgroup of $G$ and let $G' := G\/H$. Consider the map:\n\\[\n\t\\pr : M_{U, G} \\to M_{U, G'}, \\quad (X \\to Y) \\mapsto (X\/H \\to Y).\n\\]\nOne can show that for every $X' \\in M_{U, G'}$ the set $\\pr^{-1}(X')$ has a structure of a principal $M_{U, H}$-homogenous space (see e.g. proof of \\cite[Theorem 1.2]{Harbater_moduli_of_p_covers}). This structure is given by the map:\n\\[\n(X \\in M_{U, G}, \\quad Z \\in M_{U, H}) \\, \\, \\mapsto \\, \\, X_Z,\n\\]\nwhere $X : y_1^p - y_1 = f \\in k(X')$, $Z : y_2^p - y_2 = g \\in k(Y)$, $X_Z : y_3^p - y_3 = f + g$. Moreover, one proves that the map $\\pr$ has\na (non-canonical) section and thus $M_{U, G} \\cong M_{U, G'} \\times M_{U, H}$. \nThis allows one to show that $M_{U, G}$ is a direct limit of affine spaces.\\\\\n\nThe following lemma shows that (under some mild assumptions) the cover\n$X_Z \\to X'$ is at least as ramified as the cover $Z \\to Y$.\n\\begin{Lemma} \\label{lem:mQ_of_XZ}\n\tLet $X'$, $X$, $Z$ and $X_Z$ be as above.  Let also $P \\in X'(k)$ and denote by $Q$ its image on $Y$.\n\tIf $m_{Z\/Y, Q} > m_{X\/X', P} + d_{X'\/Y, P}'$ then\n\t$m_{X_Z\/X', P} > m_{Z\/Y, Q}$. \n\\end{Lemma}\n\\begin{proof}\n\tAssume that $y_1$ and $y_2$ are in local standard form at $P$ and $Q$ respectively.\n\tNote that (since $p \\nmid \\ord_Q(g)$):\n\n\t\\begin{align*}\n\t\t\\ord_P(dg\/g) &= e_{X'\/Y, P} \\cdot \\ord_Q(dg\/g) + d_{X'\/Y, P}\\\\\n\t\t&= - e_{X'\/Y, P} + d_{X'\/Y, P}.\n\t\\end{align*}\n   \n    Therefore:\n\t\\begin{equation} \\label{eqn:ord_dg}\n\t\t\\ord_P(dg) = e_{X'\/Y, P} \\cdot \\ord_Q(g) + d_{X'\/Y, P}' - 1.\n\t\\end{equation}\n\n\tThis yields:\n\n\t\\[\n\t\t\\ord_{P}(dg) = - e_{X'\/Y, P} \\cdot m_{Z\/Y, Q} + d_{X'\/Y, P}' - 1 < -m_{X\/X', P} - 1 = \\ord_{P}(df)\n\t\\]\n\n\tand $\\ord_{P}(df + dg) = \\ord_{P}(dg)$. One easily checks that~\\eqref{eqn:assumption_to_compute_mQ} holds for\n\t$g$ at $P$. Hence the result follows by Lemma~\\ref{lem:computing_mQ}.\n\\end{proof}\nIn order to prove Theorem~\\ref{thm:generic_intro} we need the following topological lemma. \n\\begin{Lemma} \\label{lem:topological}\n\tLet $\\mc X$ and $\\mc Y$ be topological spaces. Suppose that $\\mc U \\subset \\mc X \\times \\mc Y$.\n\tIf there exists a dense subset $\\mc V \\subset \\mc Y$ such that for every $y \\in \\mc V$,\n\t$\\textrm{pr}_{\\mc Y}^{-1}(y) \\cap \\mc U$ is dense in $pr_{\\mc Y}^{-1}(y)$, then $\\mc U$ is dense in $\\mc X \\times \\mc Y$.\n\\end{Lemma}\n\\begin{proof}\n\tLet $\\mc Z$ be a non-empty open subset of $\\mc X \\times \\mc Y$, we will show that $\\mc Z \\cap \\mc U \\neq \\varnothing$.\n\tNote that $\\pr_{\\mc Y}(\\mc Z)$ is a non-empty open subset of $\\mc Y$ and hence $\\pr_{\\mc Y}(\\mc Z) \\cap \\mc V \\neq \\varnothing$.\n\tLet $y \\in \\pr_{\\mc Y}(\\mc Z) \\cap \\mc V$. Then $\\mc Z \\cap \\pr_{\\mc Y}^{-1}(y)$ is a non-empty open subset\n\tof $\\pr_{\\mc Y}^{-1}(y)$. By assumption, $\\textrm{pr}_{\\mc Y}^{-1}(y) \\cap \\mc U$ is dense in $pr_{\\mc Y}^{-1}(y)$.\n\tHence:\n\n\t\\[\n\t(\\textrm{pr}_{\\mc Y}^{-1}(y) \\cap \\mc Z) \\cap (\\textrm{pr}_{\\mc Y}^{-1}(y) \\cap \\mc U) \\neq \\varnothing,\n\t\\]\n\n\twhich proves the lemma.\n\\end{proof}\n\\begin{proof}[Proof of Theorem~\\ref{thm:generic_intro}]\n\n\tDenote by $S_{U, G}$ the set of $G$-covers of $Y$ unramified over $U$ with global standard form that satisfy~\\ref{enum:A}.\n\tWe prove by induction on $\\# G$ that $S_{U, G}$ is dense in $M_{U, G}$.\n\tFor $\\# G = 1$ this is trivial. Let $G'$ and $H$ be as above.\n\tBy induction hypothesis, the set $S_{U, G'}$ is dense in $M_{U, G'}$.\n\tBy Lemma~\\ref{lem:criterion_for_gsf}\n\tfor every $X' \\in S_{U, G'}$ the covers from the set\n\n\t\\[\n\t\\mc A := \\{ X \\in \\pr^{-1}(X') : m_{X\/X', P'} > 2g_{X'} \\cdot p \\qquad \\forall_{P' \\in B' := \\textrm{ preimage of } B \\textrm{ in } X'(k)} \\} \n\t\\]\n\n\thave global standard form. Moreover, for every $X \\in \\mc A$\n\tand $P \\in X(k)$, we have $H_P = H$. Hence for every $P \\in X'(k)$, $G_P$ is the preimage of $G_{P'}'$ by the map $G \\to G'$ (where $P' \\in X'(k)$ is the image of $P$). This implies that $G_P$ is a normal subgroup of $G$ for every $P \\in X(k)$. Thus $\\mc A \\subset \\pr^{-1}(X') \\cap S_{U, G}$.\\\\\n\tOn the other hand, by Lemma~\\ref{lem:mQ_of_XZ} the set $\\mc A$ contains the set:\n\n\t\\[\n\t\\mc B := \\{ X_Z : Z \\in M_{U, H}, \\, \\, m_{Z\/Y, Q} > N_0 \\quad \\forall_{Q \\in B} \\}\n\t\\]\n\n\tfor any fixed $X \\in \\pr^{-1}(X')$ and sufficiently large $N_0$. But the set $\\{ Z \\in M_{U, H} : m_{Z\/Y, Q} > N_0 \\quad \\forall_{Q \\in B} \\}$ is dense in\n\t$M_{U, H}$. Indeed, the considered subset is contained in a complement of a finite dimensional subspace in $M_{U, H}$. Thus the proof follows by Lemma~\\ref{lem:topological}.\n\\end{proof}\n\n\\subsection{Example: Heisenberg group covers}\nConsider the Heisenberg group $\\textrm{mod } p$:\n\\[\nE(p^3) = \\langle a, b, c : a^p = b^p = c^p = e, c = [a, b], c \\in Z(E(p^3)) \\rangle.\n\\]\nLet $Y$ be a smooth projective curve over $k$ and let $f_1, f_2, f_3 \\in k(Y)$.\nDefine $X$ to be a smooth projective curve with the function field $k(Y)(y_1, y_2, y_3)$, where:\n\\begin{equation} \\label{eqn:Ep3cover}\n\\begin{aligned} \n\ty_1^p - y_1 &= f_1,\\\\\n\ty_2^p - y_2 &= f_2,\\\\\n\ty_3^p - y_3 &= f_3 + f_2 \\cdot (y_1 + y_2).\n\\end{aligned}\t\n\\end{equation}\nThen $E(p^3)$ acts on $X$ via the formulas:\n\\begin{align*}\n\ta^*(y_1, y_2, y_3) &= (y_1 + 1, y_2, y_3 + y_2)\\\\\n\tb^*(y_1, y_2, y_3) &= (y_1 + 1, y_2 + 1, y_3)\\\\\n\tc^*(y_1, y_2, y_3) &= (y_1, y_2, y_3 - 1).\n\\end{align*}\nIt turns out that every $E(p^3)$-cover of $Y$ is of this form (this follows from results of Saltman, cf.~\\cite{Saltman_Noncrossed_product}).\nIn order to construct examples of covers with magical elements, it is useful to\nestimate genus of a curve in a tower of covers. If $Z \\to Y$ is\nan Artin--Schreier cover with the equation~\\eqref{eqn:artin_schreier} then Riemann--Hurwitz formula yields:\n\\begin{equation} \\label{eqn:genus_AS}\n\tg_Z < p \\cdot (g_Y + \\deg_Y f),\n\\end{equation}\nwhere $\\deg_Y f$ is the degree of $f$ treated as a morphism $Y \\to \\PP^1$.\n\\begin{Corollary} \\label{cor:example_from_intro}\n\tLet $Y$ and $X$ be as in Example~\\ref{ex:intro}. Then $X \\to Y$ is an $E(p^3)$-cover satisfying the conditions~\\ref{enum:A} and \\ref{enum:B}. Moreover:\n\n\t\\[\n\tG_{Q_1} \\cong E(p^3), \\quad G_{Q_2} \\cong \\ZZ\/p \\times \\ZZ\/p, \\quad G_{Q_3} \\cong \\ZZ\/p,\n\t\\]\n\n\twhere $Q_i := (\\alpha_i, 0) \\in Y(k)$.\n\\end{Corollary}\n\\begin{proof}\n\tNote that $X$ is defined by the equations~\\eqref{eqn:Ep3cover},\n\twhere: \n\n\t\\begin{align*}\n\t\tf_1 &= (x-\\alpha_1)^{-a_1},\\\\ \n\t\tf_2 &= (x-\\alpha_1)^{-a_2} \\cdot (x-\\alpha_1)^{-b_2},\\\\\n\t\tf_3 &= (x-\\alpha_1)^{-a_3} \\cdot (x-\\alpha_1)^{-b_3} \\cdot (x-\\alpha_1)^{-c_3}.\n\t\\end{align*}\n\n\tHence $X \\to Y$ is an $E(p^3)$-cover. Let $X_1$ be the $\\ZZ\/p$-cover of $Y$, given by $y_1^p - y_1 = f_1$\n\tand let $X_2$ be the $\\ZZ\/p \\times \\ZZ\/p$-cover of $Y$, given by:\n\n\t\\[\n\ty_1^p - y_1 = f_1, \\quad y_2^p - y_2 = f_2.\n\t\\]\n\n\tNote that\n\n\t\\[\n\tm_{X_1\/Y, Q_1} = 2a_1 > 2g_Y \\cdot p\n\t\\]\n\n\tand hence $X_1 \\to Y$ has a global standard form by Lemma~\\ref{lem:criterion_for_gsf}.\n\tBy~\\eqref{eqn:genus_AS}, $g_{X_1} < p \\cdot (1 + 2a_1) < 3a_1 \\cdot p$.\n\tOne checks that $f_2$ satisfies the condition~\\eqref{eqn:assumption_to_compute_mQ}.\n\tHence, using Lemma~\\ref{lem:computing_mQ} and~\\eqref{eqn:ord_dg},\n\tfor $P \\in X_1(k)$ in the preimage of $Q_1$:\n\n\t\\begin{align*}\n\t\tm_{X_2\/X_1, P} &= -\\ord_P(df_2) - 1\\\\\n\t\t&= p \\cdot 2a_2 - d_{X_1\/Y, P}' - 1\\\\\n\t\t&= p \\cdot 2a_2 - (p-1) \\cdot 2 a_1 - 1\\\\\n\t\t&> 2g_{X_1} \\cdot p. \n\t\\end{align*}\n\n\tHence, by Lemma~\\ref{lem:criterion_for_gsf}, $X_2 \\to X_1$ also has a global standard form.\n\tNote now that by~\\eqref{eqn:genus_AS}:\n\n\t\\begin{align*}\n\t\tg_{X_2} &< p \\cdot (g_{X_1} + \\deg_{X_1} f_2) < 3p^2 \\cdot (a_1 + a_2 + b_2),\\\\\n\t\td'_{X_2\/Y, Q_1} &< 2p^2 \\cdot (a_1 + a_2).\n\t\\end{align*}\n\n\tTherefore for any $P \\in X_2(k)$ in the preimage of $Q_1$:\n\n\t\\begin{align*}\n\t\t-\\ord_P(df_3) &> p^2 \\cdot 2a_3 - 2p^2 \\cdot (a_1 + a_2).\n\t\\end{align*}\n\n\tOne easily checks that~\\eqref{eqn:assumption_to_compute_mQ} is satisfied for\n\t$f_3 + (y_1 + y_2) \\cdot f_2$ and that $\\ord_P(d(f_3 + (y_1 + y_2) \\cdot f_2)) = \\ord_P(df_3)$.\n\tHence:\n\n\t\\begin{align*}\n\t\tm_{X\/X_2, P} = -\\ord_P(d(f_3 + (y_1 + y_2) \\cdot f_2)) - 1\n\t\t> 2g_{X_2} \\cdot p.\n\t\\end{align*}\n\n\tTherefore $X \\to Y$ has a global standard form. Moreover, one checks that $m_{X\/X_2, P} > 0$ for $P \\in X(k)$ in the preimage of $Q_2$. Hence $G_{P} = \\ZZ\/p \\times \\ZZ\/p$ (since $X \\to X_1$ is totally ramified over $P$ and $X_1 \\to Y$ is unramified\n\tover $Q_2$). Analogously, $G_{Q_3} = \\ZZ\/p$. This finishes the proof. \n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction to the model}\n\\label{sec:introduction}\n\nThe Standard Model has some well-known observational shortcomings: it\n\\begin{inparaenum}[\\it (a)]\n    \\item does not contain neutrino masses at the renormalizable level\n    \\item cannot explain the observed baryon asymmetry of the Universe (BAU)\n    \\item cannot explain dark matter.\n\\end{inparaenum}\nOne of the many possible solutions to these problems consists in adding to the Standard Model two or more right-handed neutrinos $N_I$ --- or \\emph{Heavy Neutral Leptons} (see e.g.\\ ref.~\\cite{FIPs}).\nAs Standard Model singlets (i.e.\\ completely neutral particles), they admit a Majorana mass term which, combined with the Yukawa interaction, produces a non-diagonal mass term after electroweak symmetry breaking. This leads to mixing between the neutrino flavor states $\\nu_{\\alpha}$ and the new heavy mass eigenstates $N_I$, which thus behave as heavy Majorana neutrinos with interactions suppressed by a small mixing angle $\\Theta_{\\alpha I}$:\n\\begin{equation*}\n    \\nu_\\alpha \\approx V_{\\alpha i}^{\\mathrm{PMNS}} \\nu_i + \\Theta_{\\alpha I} N_I^c.\n\\end{equation*}\n\n\\section{ATLAS constraints on HNLs}\n\\label{sec:constraints}\n\nHNLs have elicited a strong interest from the experimental community. Here we focus on a specific search~\\cite{Aad:2019kiz} by the ATLAS collaboration, for HNLs in the mass range $M_N \\in [5,50]\\,\\si{GeV}$, produced in $W$ decays and decaying promptly to the trilepton final states $e^{\\pm} e^{\\pm} \\mu^{\\mp}$ (electron channel) and $\\mu^{\\pm} \\mu^{\\pm} e^{\\mp}$ (muon channel) plus missing transverse energy. Both channels have contributions from both lepton number conserving (LNC) and lepton number violating (LNV) processes.\nLike most experiments, ATLAS has reported their limits for simplified models only, where a single Majorana HNL mixes with either the electron or muon neutrino, but not both. The LNC processes depend on both mixing angles, therefore their contribution was not included in this original interpretation.\n\n\\section{Parameter space of the model}\n\\label{sec:parameter_space}\n\nThe seesaw mechanism, being responsible for the generation of neutrino masses, relates the HNL masses and mixing angles to the measured neutrino oscillation parameters~\\cite{NuFIT5.0}.\nIn addition, if HNLs have roughly the same interaction strengths and are within experimental reach, then it can be shown that their masses must be nearly degenerate \\cite{Kersten:2007vk,Drewes:2019byd}.\nIn what follows, we will focus on a minimal seesaw model with only two nearly degenerate HNLs.\n\n\\paragraph{Constraints on the mixing angles}\n\nFrom the point of view of collider experiments, a pair of nearly degenerate HNLs will behave as a single particle. Combining this degeneracy with the seesaw formula and neutrino oscillation data~\\cite{NuFIT5.0}, we obtain a constraint on the allowed ratios of squared mixing angles with the electron, muon and tau flavors, as shown in \\cref{fig:ternary_plot} \\cite{Drewes:2016jae,Caputo:2017pit}.\nThe original interpretation set constraints on only two points in this plane: the right (electron channel) and top (muon channel) vertices of this triangle, which can be seen to be incompatible with neutrino oscillation data within the model under consideration.\\footnote{These constraints would be significantly relaxed by the addition of extra nearly-degenerate HNLs.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.76\\textwidth]{plots\/ternary_plot_SKoff_with_benchmarks.pdf}\n    \\caption{Allowed ratios of the three squared mixing angles $|\\Theta_{eI}|^2$, $|\\Theta_{\\mu I}|^2$ and $|\\Theta_{\\tau I}|^2$.}\n    \\label{fig:ternary_plot}\n\\end{figure}\n\n\\paragraph{HNL oscillations}\n\nIn addition, nearly degenerate HNLs can undergo coherent oscillations~\\cite{Tastet:2019nqj}, i.e.\\ a periodic modulation of their decay rate (with opposite phases for LNC and LNV processes) as a function of the proper time $\\tau = \\sqrt{(\\smash[b]{x_{\\mathrm{decay}}-x_{\\mathrm{prod}}})^2}$ between the HNL production and decay, with the oscillation (angular) frequency given by the mass splitting $\\delta M$ between the two mass eigenstates (as represented in \\cref{fig:oscillations}).\nWe will focus on the two extreme cases:\n\\begin{inparaenum}[\\it (a)]\n    \\item Dirac-like HNLs (observed before the onset of oscillations, see \\cref{fig:osc_Dirac-like}) for which the rate of LNV processes is suppressed compared to LNC, and\n    \\item Majorana-like HNLs (observed after many oscillations, see \\cref{fig:osc_Majorana-like}) for which the integrated rates\\footnote{The differential distribution of the decay products will differ between LNC and LNV due to spin correlations \\cite{Tastet:2019nqj}.} of LNC and LNV processes are the same.\n\\end{inparaenum}\nSince the rates of LNC processes vanish under the single-flavor assumption, the original analysis was only sensitive to Majorana-like HNLs.\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.33\\textwidth}\n        \\centering\n        {\\hspace*{2em}\\small($\\delta M\/\\Gamma = \\pi\/10$)}\\\\\n        \\includegraphics[width=\\linewidth]{plots\/oscillations_Dirac.pdf}\n        \\caption{Dirac-like}\n        \\label{fig:osc_Dirac-like}\n    \\end{subfigure}\\hfill%\n        \\begin{subfigure}[b]{0.33\\textwidth}\n        \\centering\n        {\\hspace*{2em}\\small($\\delta M\/\\Gamma = \\pi$)}\\\\\n        \\includegraphics[width=\\linewidth]{plots\/oscillations_resolvable.pdf}\n        \\caption{Visible oscillations}\n    \\end{subfigure}\\hfill%\n        \\begin{subfigure}[b]{0.33\\textwidth}\n        \\centering\n        {\\hspace*{2em}\\small($\\delta M\/\\Gamma = 10\\pi$)}\\\\\n        \\includegraphics[width=\\linewidth]{plots\/oscillations_Majorana.pdf}\n        \\caption{Majorana-like}\n        \\label{fig:osc_Majorana-like}\n    \\end{subfigure}\n    \\caption{HNL oscillations in three different regimes ($\\Gamma$ is the total HNL width).}\n    \\label{fig:oscillations}\n\\end{figure}\n\n\\section{Findings from the reinterpretation}\n\\label{sec:reinterpretation}\n\nOur reinterpretation method is described in details in ref.~\\cite{Tastet:2021vwp}. \\Cref{fig:sketch} attempts to briefly summarize its main features. We vary the HNL mass and mixing angles, and solve for $\\mathrm{CL}_s = 0.05$ (using a simplified background model) in order to obtain the recast exclusion limit. We perform a scan for each neutrino mass hierarchy and for both Dirac- and Majorana-like HNLs. To more easily visualize the scan over the mixing angles, we define a number of \\emph{benchmark points} (visible in \\cref{fig:ternary_plot}) which represent both typical and extreme ratios of the squared mixing angles. In order to consistently compare different benchmarks, we express the recast limits in terms of the total mixing angle $U_{\\mathrm{tot}}^2$ (summed over all three flavors and the two mass eigenstates). We finally compute a conservative bound by marginalizing over all the allowed ratios of squared mixing angles.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{plots\/reinterpretation_workflow_rasterized.pdf}\n    \\caption{Sketch of the reinterpretation workflow (input and output parameters are in red).}\n    \\label{fig:sketch}\n\\end{figure}\n\n\\paragraph{Majorana-like HNLs}\n\nWe obtain the recast limits shown in \\cref{fig:results_Majorana}, expressed as a function of the HNL mass $M_N$ and its total mixing angle $U_{\\mathrm{tot}}^2$. The black lines correspond to the simplified models originally used by ATLAS (recomputed for consistency), while the numbered colored lines denote the recast limits for the various benchmarks. We see that the recast limits can be more than an order of magnitude weaker than the original ones, with the worst case corresponding to tau flavor dominance (where the branching ratios into channels involving $\\tau$ leptons are increased at the expense of the two search channels, as already observed in ref.~\\cite{Abada:2018sfh} in the context of displaced searches). The blue area shows the lower and upper bounds for the recast limits when scanning over all the allowed mixing angles, and by extension the gray area is conservatively excluded within this two-HNL model.\n\n\\paragraph{Dirac-like HNLs}\n\nThe recast limits are shown in \\cref{fig:results_Dirac}. Unlike in the single-flavor case where all LNC cross-sections vanish, we can now set a conservative limit thanks to the constraints from neutrino oscillations, which forbid trivial ratios of the mixing angles within this model. However, for all benchmarks, the recast limits are weaker than those obtained for a single Majorana HNL mixing with a single flavor (gray lines), by up to three orders of magnitude. The weakest limits are obtained when the electron or muon mixing angle is much smaller than the two others.\n\n\\begin{figure}\n    \\begin{subfigure}[b]{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{plots\/recast_limits_total_U2_Majorana_NH.pdf}\n        \\caption{Majorana-like}\n        \\label{fig:results_Majorana}\n    \\end{subfigure}\\hfill%\n    \\begin{subfigure}[b]{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=0.95\\linewidth]{plots\/recast_limits_total_U2_Dirac_NH.pdf}\n        \\caption{Dirac-like}\n        \\label{fig:results_Dirac}\n    \\end{subfigure}\n    \\caption{Recast limits (taking the NH as an example, the IH is similar; mind the different $y$-axes).}\n\\end{figure}%\n\n\\section{Lessons learned \\& recommendations for experiments}\n\\label{sec:conclusion}\n\nThese results show why the limits reported for simplified benchmarks \\emph{should not be used directly} (e.g. by equating the $U_{\\mathrm{tot}}^2$) to experimentally test more realistic models. Instead, they must be \\emph{reinterpreted} within those models. Otherwise, we incur the risk of \\emph{wrongly excluding} valid models or regions in parameter space.\n\nPerforming an accurate reinterpretation is a non-trivial task. In particular, computing the signal efficiencies and modeling the background can be difficult, even with a good knowledge of the experiment. To help with the former, we propose in ref.~\\cite{Tastet:2021vwp} a reweighting method that allows one to exactly extrapolate the expected signal to any combination of mixing angles, using only a handful of constants that can be easily computed (and published) by experiments.\nA similar method could easily be devised for other models of feebly interacting particles.\n\nFinally, since accurately modeling the background is extremely difficult --- if not impossible --- for people working outside the experimental collaboration, a pragmatic solution that would allow theorists to reinterpret the results within their favorite model or set of parameters would be to release either\n\\begin{inparaenum}[\\it (a)]\n    \\item the full likelihood, as working code, or\n    \\item a simplified likelihood or\n    \\item the covariance matrix between all background bin counts in all channels.\n\\end{inparaenum}\nThis is in line with the recommendations from the LHC Reinterpretation Forum~\\cite{Abdallah:2020pec}.\n\n\\paragraph{Acknowledgments}\n\nThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (GA 336581, 694896) and under the Marie Skolodowska-Curie grant agreement No 860881-HIDDeN; from the Carlsberg foundation; from the Swiss National Science Foundation Excellence under grant 200020B\\underline{ }182864; from the Spanish MINECO through the Centro de excelencia Severo Ochoa Program under grant SEV-2016-0597; and from the Spanish \"Agencia Estatal de Investigac\u00edon\"(AEI) and the EU \"Fondo Europeo de Desarrollo Regional\" (FEDER) through the project PID2019-108892RB-I00\/AEI\/10.13039\/501100011033.\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\part{\\partial_t}\n\\def\\partial_s}\\def\\parz{\\partial_z{\\partial_s}\\def\\parz{\\partial_z}\n\\def{d\\over{dt}}{{d\\over{dt}}}\n\\def{d\\over{ds}}{{d\\over{ds}}}\n\\def\\i#1#2{\\int_{#1}^{#2}}\n\\def\\int_{S^n}}\\def\\wtilde{\\widetilde{\\int_{S^n}}\\def\\wtilde{\\widetilde}\n\\def\\widehat{\\widehat}\n\\def\\R^n}\\def\\B{{\\cal B}{{\\hbox{\\mathbold\\char82}}^n}\\def\\B{{\\cal B}}\n\\def\\int_M{\\int_M}\n\\def\\int_{\\sR^n}}\\def\\bSn{{\\bS}^n{\\int_{{\\hbox{\\smathbold\\char82}}}\\def\\mR{{\\hbox{\\mmathbold\\char82}}^n}}\\def\\bSn{{{\\hbox{\\tenbi\\char83}}}^n}\n\\def\\ref#1{{\\bf{[#1]}}}\n\\font\\twelverm=cmr12 scaled \\magstep2\n \\def\\,{\\hbox{\\it CON}}(\\Rn)}\\def\\consn{\\,{\\hbox{\\it CON}}(S^n){\\,{\\hbox{\\it CON}}(\\R^n}\\def\\B{{\\cal B})}\\def\\consn{\\,{\\hbox{\\it CON}}(S^n)}\n\\def\\grn{{g_{\\hskip-1.5pt {{\\hbox{\\sixbi\\char82}}^{\\smallfivrm\\char110}}}^{}}}\\def\\gsn{{g_{\\hskip-1.5pt \n{\\scriptscriptstyle{S^n}}}^{}}}\n\\defw^{{1\\over2}}\\lapdi w^{{1\\over2}}{w^{{1\\over2}}\\lap^{\\!-{d\/2}} w^{{1\\over2}}}\n\\defW^{{1\\over2}}A_d^{-1} W^{{1\\over2}}{W^{{1\\over2}}A_d^{-1} W^{{1\\over2}}}\n\\def{\\rm {ess\\hskip.2em sup}}{{\\rm {ess\\hskip.2em sup}}}\n\\def\\partial{\\partial}\n\\def\\omega_{2n+1}{\\omega_{2n+1}}\n\\def{\\rm {ess\\hskip2pt inf }}\\;{{\\rm {ess\\hskip2pt inf }}\\;}\n\n\\def\\int_\\Sigma{\\int_\\Sigma}\n\\def{\\rho_H^{}}{{\\rho_H^{}}}\n\n\\def{\\ts{n\\over2}}}\\def\\Qhalf{{\\tsyle{Q\\over2}}}\\def\\half{{\\ts{1\\over2}}{{\\ts{n\\over2}}}\\def\\Qhalf{{\\tsyle{Q\\over2}}}\\def\\half{{\\ts{1\\over2}}}\n\\def{\\cal K}}\\def\\A{{\\bf A}}\\def\\D{{\\cal D}}\\def\\G{{\\cal G}}\\def\\RR{{\\cal R}{{\\cal K}}\\def\\A{{\\bf A}}\\def\\D{{\\cal D}}\\def\\G{{\\cal G}}\\def\\RR{{\\cal R}}\n\\def{\\cal W}{{\\cal W}}\n\n\\def{\\cal I}} \\def{\\cal F}{{\\cal F}}\\def\\cR{{\\cal R}{{\\cal I}} \\def{\\cal F}{{\\cal F}}\\def\\cR{{\\cal R}}\n\\def\\wtilde k_m{\\wtilde k_m}\n\n\\def\\dot\\times}\\def\\op{{\\hbox { Op}}}\\def\\J{{\\cal J}{\\dot\\times}\\def\\op{{\\hbox { Op}}}\\def\\J{{\\cal J}}\n\\def{\\cal T}{{\\cal T}}\n\\def \\L{{\\bf L}} \n\n\n\\def\\rmi{{\\rm {(i) }}}\\def\\ii{{\\rm {(ii) }}}\\def\\iii{{\\rm (iii) }}\\def\\iv{{\\rm \n{(iv) }}}\n\n\\def\\ker{{\\rm Ker}} \n\\def{\\rm {(v) }}}\\def\\vi{{\\rm {(vi) }}{{\\rm {(v) }}}\\def\\vi{{\\rm {(vi) }}}\n\n\n\n\\def\\Gamma\\big({Q\\over2}\\big){\\Gamma\\big({Q\\over2}\\big)}\n\\def{\\wtilde\\Theta}{{\\wtilde\\Theta}}\n\\def{\\cal H}{{\\cal H}}\n\n \n\n\\centerline{\\bf{  Adams inequalities on measure spaces}}\n\\bigskip\\centerline{Luigi Fontana, Carlo Morpurgo}\n\\vskip1em\n\\midinsert\n{\\smalsmalbf Abstract. }{\\smallfonts In 1988 Adams obtained sharp Moser-Trudinger inequalities on bounded domains of $\\mR^n$. The main step was a sharp exponential integral inequality for convolutions with the Riesz potential. In this paper we extend and improve  Adams' results to  functions defined on arbitrary measure spaces with finite measure. The Riesz fractional integral is replaced by  general integral operators, whose kernels satisfy suitable and explicit growth conditions, given in terms of their distribution functions; natural conditions for sharpness are also given. Most of the known results about Moser-Trudinger inequalities can be easily adapted to our unified scheme. We give some new applications of our theorems, including:   sharp higher order Moser-Trudinger trace inequalities, sharp Adams\/Moser-Trudinger inequalities for general elliptic differential operators (scalar and vector-valued), for  sums of weighted potentials, and for operators  in the CR setting. \n}\n\n\\endinsert\\bigskip\n\\centerline{\\bf INTRODUCTION}\\bigskip\\bigskip\n\n\n\nExponential integrability can often compensate for  lack of boundedness, as a natural (although  weaker) condition. There are \nnumerous  important instances of this idea in the literature, the first is perhaps due to Zygmund. It is well known that the conjugate function of a bounded function on the torus $T$ need not  be bounded, but in 1929 Zygmund proved that for all $\\lambda < {{\\pi}\\over{2}}$ the conjugate function $\\tilde f$  satisfies\n$$\n\\int_{{\\bf T}} \\exp\\Big({ \\lambda |\\tilde f(\\theta)|}\\Big) d\\theta \\leq C_{\\lambda}\n$$\nwhenever $f$ is real-valued and  belongs to the closed unit ball of $L^{\\infty}({\\bf T})$ ([Z], Ch. VII). Cancellation, through Cauchy's integral formula, plays the central role in the proof of this result.\n\nOn the other hand, size has the major role in  the chain of results that followed a 1967 paper by Trudinger, in which he showed that exponential integrability fills the gap in Sobolev's immersion\ntheorem:\n\n\n\n\\proclaim {Theorem [Tr]}. Let $\\Omega $ be a bounded region in ${{\\hbox{\\mathbold\\char82}}}^n$, $n>1$.  \nThere exist constants $\\lambda $ and $C$ such that, if $u$ satisfies the Dirichlet condition on\n$\\partial \\Omega $ and \n$(\\int_{\\Omega}|\\nabla u|^{n})^{{1\\over n}} \\leq 1$, then \n$$\n\\int_{\\Omega} \\exp\\Big({\\lambda |u|^{{{n}\\over{n-1}}}}\\Big) dx \\leq C.\\eqno(1)\n$$\n\\par\nIn 1971 Moser sharpened the result by showing that\n$\n\\lambda = n \\omega _{n-1}^{{{1}\\over{n-1}}}\n$\nis  best possible in (1), where  $\\omega_{n-1}$\nis the surface measure of the unit sphere in ${{\\hbox{\\mathbold\\char82}}}^{n}$.\n\\eject\nDue to the wide range of applications in PDE's, Differential Geometry  and String Theory, Moser's result triggered an enormous amount of work in the years that followed, and  up to present time. Several aspects and extensions of Moser's inequality were studied, and   still are part of an active field of research: existence of extremals, Neumann conditions rather than Dirichlet,     \nsettings other than $\\R^n}\\def\\B{{\\cal B}$, higher order derivatives, and more. All but a handful of the references listed in the back of this article \ndeal with Moser-Trudinger inequalities, in some form or another, and the list is only partial.\n\n\nAdams' paper in 1988, however, represents a true turning point. Not only did he extend\nMoser's sharp result to higher order derivatives, but he also set the strategy that opened the way to most of the later work in the field. We recall here the basic developments. Adams' generalization of\nMoser's theorem is\n\n\\bigskip\n\n\\proclaim Theorem [Ad1].\nLet $\\Omega $ be a domain  in ${{\\hbox{\\mathbold\\char82}}}^{n}$ and  $m$ be a positive integer \nstrictly smaller than $n$. There are constants $\\beta(m,n)$ and $C$ with the following\nproperty: If $u \\in C^{m}({{\\hbox{\\mathbold\\char82}}}^{n})$\nhas support contained in $\\Omega$ and\n$\\parallel \\nabla^{m}u \\parallel_{n\/m}  \\leq 1$, then\n$$\n\\int_{\\Omega} \\exp\\Big[{\\beta(m,n) |u(x)|^{{{n}\\over{n-m}}}}\\Big] dx \\leq C.\\eqno(2)\n$$\n\\par\nThe constant $\\beta(m,n)$ is given explicitly in [Ad1] and it  is  sharp (see also Theorem 6 in section 5). Also, $\\nabla^{m}$ means $\\Delta^{{m \\over 2}}$\nwhen $m$ is even and $\\nabla \\Delta^{{{m-1}\\over {2}}}$ when $m$ is odd, where $\\Delta$ denotes the positive Laplacian on $\\R^n}\\def\\B{{\\cal B}$.\n              \\smallskip                                                   \nAdams' approach consists of five main steps. \\smallskip\n\\noindent{\\bf Step 1.} Represent  $u$ in terms of $\\nabla^m u$,  via  convolutions with the Riesz potential. \n\\smallskip\\noindent{\\bf Step 2.} Formulate the following sharp theorem on exponential integrability for Riesz potentials (a dual, but more general, version of the above theorem that has its own relevance).\nThe first theorem follows immediately from the second, apart for some\nextra work necessary to ensure that the inequality is indeed sharp.\n\n\\bigskip\n\n\\proclaim  Theorem [Ad1].\nFor $1 < p < \\infty$ , there is a constant $C$ such that for all \n$f \\in L^{p} ({{\\hbox{\\mathbold\\char82}}}^{n})$ with support contained in $\\Omega $ and $\\parallel f \\parallel_{p} \\leq 1 $, \n$$\n\\int_{\\Omega} \\exp\\Big[{{{n}\\over {\\omega_{n-1}}}|I_{\\alpha} \\ast f(x)|^{p'} } \n \\Big]dx \\leq C\\eqno(3)\n$$\nwhere $\\alpha = n\/p $, $1\/p + 1\/p' = 1$, and \n$I_{\\alpha} \\ast f(x) = \\int | x-y |^{\\alpha-n} f(x) dy$.\nThe constant $n\/\\omega_{n-1}$ cannot be replaced by any larger number without  \nforcing  $c_{0}$  to depend on $f$ as well as on $p$ and $n$.\n\\par \n\\eject \n\\smallskip\\noindent{\\bf Step 3.}  The third  step of Adam's strategy  is to reduce the proof of the above theorem to a one dimensional exponential inequality by using  a lemma due to O'Neil: if $T$ is a convolution operator on a measure space, then \n$$T(f,g)^{**}(t)\\le t f^{**}(t)g^{**}(t)+\\int_t^\\infty f^*(u)g^*(u)du,\\qquad t>0 \\eqno(4)$$\nwhere $f^*$ denotes nonincreasing rearrangement on the half-line, and $f^{**}(t)=t^{-1}\\displaystyle\\int_0^t f^*(u) du$.\n\n\\bigskip\n\\smallskip\\noindent{\\bf Step 4.} The next step is to prove the one dimensional exponential inequality derived in  step 3 by means of a technical lemma,  now known as the ``Adams-Garsia's Lemma\". \n\\medskip\\noindent{\\bf Step 5.} The final step is to show that the exponential constant is sharp, by showing that for any larger constant one can find a suitable sequence of functions that makes the exponential integral arbitrarily large.\n  \\medskip\nIn his PhD thesis (1991) Fontana adapted Adams' strategy, and extended his results  in the setting of compact Riemannian manifolds [F]. In that situation the Green function  replaces the Riesz potential in Step 1; the corresponding integral representation is no longer a global convolution,  but locally the Green kernel is a  perturbed Riesz potential. These facts eventually lead to  suitable versions of  O'Neil's lemma  and Adams-Garsia's lemma; these  modified lemmas  could not be deduced from the original ones, even though the original proofs were successfully adapted to the perturbed setting [F].\n\n Several other authors also  used Adams strategy, sometimes partially,  in order to prove sharp Moser-Trudinger estimates in various  settings. In most cases, like in [F], some    individual steps had to be adapted, and sometimes their proofs were only sketched, or even omitted. \n \n\nRecently ([BFM]), the authors of this paper, in joint project with Tom Branson, needed a sharp form of various Moser-Trudinger inequalities in the CR setting in order to obtain the sharp version of Beckner-Onofri's inequality on the complex  sphere. Independently Cohn and Lu [CoLu 1,2] had  worked out Adams and Moser-Trudinger sharp estimates\nin some very special cases, which were not suitable to our needs. While working out yet another version of Adams strategy, we realized that steps 2,3,4,5 could be formulated in an arbitrary measure space, for integral operators  more general than convolutions, and with kernels  satisfying suitable growth  and integral conditions.\n \n\n It was then that we seriously looked into the possibility of a general result that would encompass and unify the various  Adams-type  procedures,  with the hope that it would prove to be useful to authors in need of such sharp estimates in a variety of situations. Stripped down to its essence, the present paper could be summarized as follows. \n\n\\eject\n\nSuppose that $T$ is an integral operator of type\n$$Tf(x)=\\int_M k(x,y)f(y)d\\mu(y)\\,,\\qquad x\\in N$$\nwhere $(M,\\mu)$,$\\,(N,\\nu)$ are  measure spaces with finite measure, and suppose that the kernel $k(x,y) $ satisfies\n$$\\sup_{x\\in N}\\mu\\Big(\\{y\\in M: \\,|k(x,y)|>s\\}\\Big)\\le A s^{-\\beta}\\Big(1+O(\\log^{-\\gamma} s)\\Big)\\eqno(5)$$\n$$\\sup_{y\\in M}\\nu\\Big(\\{x\\in N:\\, |k(x,y)|>s\\}\\Big)\\le B s^{-\\beta_0}\\eqno(6)$$\nas $s\\to+\\infty$, where $\\beta>1$, $\\beta'$ is the conjugate exponent, $0<\\beta_0\\le \\beta$, and $\\gamma>1$.\n Then we have an exponential inequality\nof type \n$$\\int_N \\exp\\bigg[{\\beta_0\\over A\\beta} \\bigg({|Tf(x)|\\over\\|f\\|_{\\beta'}}\\bigg)^{\\beta}\\bigg]\\,d\\nu(x)\\le C\\eqno(7)$$\nfor any $f\\in L^{\\beta'}(M)$.\nAs for the sharpness statement, if equality holds in (5) then the constant $\\beta_0\/(A\\beta)$ in (7) is sharp, provided that \ncertain reasonable ``regularity\" conditions are satisfied by the kernel $k$.\n\n The main feature of this result, which is Theorem 1 in the next section, is   that it reduces Moser-Trudinger inequalities \nfor integral operators, or in ``dual form\", to a couple of  estimates for the distribution functions of their kernels, and the sharpness result (under suitable but reasonable geometric conditions) to a single integral estimate (see d) in Theorem 4). \nIn some cases estimates (5) and (6) are rather trivial to check, like for the Riesz potential $k(x,y)=|x-y|^{d-n}$, on a domain $\\Omega$, for which\n$$\\sup_{x\\in\\Omega}| \\{y\\in\\Omega :\\,|x-y|^{d-n}>s\\}|={\\omega_{n-1}\\over n}\\,s^{-{n\\over n-d}}.\\eqno(8)$$\nIn other situations the  asymptotics of the distribution function of $k$ could be a bit more involved, but they are usually  a consequence of an  asymptotic expansion of the kernel $k$ around its singularity. For example, kernels that satisfy (5) and (6) are those of type\n$$k(x,y)=c(d,n)|x-y|^{d-n}+O(|x-y|^{d-n+\\epsilon}\\def\\part{\\partial_t})\\eqno(9)$$\nsome $\\epsilon}\\def\\part{\\partial_t>0$, or more generally of type  \n$$k(x,y)=k_{d-n}(x,x-y)+O(|x-y|^{d-n+\\epsilon}\\def\\part{\\partial_t})\\eqno(10)$$\nsome suitable $k_{d-n}(x,z)$ homogeneous of order $d-n$ in $z$ (see Lemma 9).\nThese are in fact more than just examples. It was already shown  by Fontana in [F]  that  one can still set up the Adams machinery for powers of Laplace-Beltrami operators on compact manifolds without boundary,  even though such operators have  fundamental solutions  that  do not satisfy the precise identity  (8), but instead satisfy a perturbed version like (9), in local coordinates. \n \nThe fact that error terms are allowed in the asymptotics of the kernels or their distribution functions is an important point of our theory.   Indeed,   (10) is precisely the type of expansion satisfied by \nthe classical parametrix   of elliptic pseudodifferential operators of order $d$  on bounded domains of $\\R^n}\\def\\B{{\\cal B}$ (or on compact manifolds, in local coordinates). Whenever an elliptic operator $P$, say on a domain $\\Omega$, has a fundamental solution $T$ with such kernel, we can  write any compactly supported smooth function as $u=T(Pu)$ and almost  immediately obtain a sharp Moser-Trudinger inequality of type \n$$\\int_\\Omega \\exp\\bigg[A^{-1} \\bigg({|u(x)|\\over\\|Pu\\|_{p}}\\bigg)^{p'}\\bigg]\\,dx\\le C$$\nwhere the sharp constant $A^{-1}$ depends on the principal symbol of $P$. This is in fact one of the applications we give of our main Theorem, extending Adams original result (2) to  a wide class of  scalar and vector-valued elliptic differential operators (see Theorems 10, and 12). In the special case of second-order elliptic operators, the sharp constant is even more explicitly described in terms of the  matrix  formed by the second order coefficients (see Corollary 11).\n\n\nAnother feature of our main theorem  is that it offers ample flexibility in the choice of the base measure spaces $(M,\\mu)$ and $(N,\\nu)$. To illustrate this point we offer an  extension of a very recent result  of Cianchi [Ci1] who proved that if $\\nu$ is a Borel measure on $\\Omega\\subseteq {\\hbox{\\mathbold\\char82}}^n$ satisfying $\\nu\\big(B(x,r)\\cap \\Omega\\big)\\le C r^\\lambda$, for suitable $\\lambda\\in (0,n]$, and for small $r$, then \n$$\\int_\\Omega \\exp\\bigg[\\lambda \\omega_{n-1}^{1\\over n-1} \\bigg({|u(x)|\\over\\|\\nabla u\\|_{n}}\\bigg)^{n'}\\bigg]\\,d\\nu(x)\\le C\\eqno(11)$$\nfor all $u\\in W_0^{1,n}(\\Omega)$. As Cianchi observed, this result immediately leads to inequality for traces of functions, either   on boundaries of smooth $\\lambda$-dimensional submanifolds of $\\R^n}\\def\\B{{\\cal B}$ or on sets of fractal type.  Cianchi's proof of the above inequality did not follow the representation formula as in Adams' original paper, step 1. By  use of a  trace Sobolev inequality also due to Adams (see (59)) and clever rearrangement results, Cianchi is however able to  make some contact with Adams' original steps 2,3,4,5.\n\nIn Theorems 6 and 7, we extend (11) to higher order operators and potentials. We especially hope to show how (11) and its higher order versions  are  part of the same large family of Adams\/Moser-Trudinger inequalities, and are  in fact simple applications of our main theorems. The role of the constant is clearly explained in terms of the interactions between the base measures $d\\nu(x), \\,d\\mu(y)=dy, $ and the Riesz potentials, as given in (5) and (6).\n\nWe would like to point out that our original formulation of (5)-(7) had $\\beta=\\beta_0$, and was based (among many other things)  on an improved version of O'Neil's lemma given as in (20). It was only after being aware of Cianchi's result that we started looking for a further improvement of O'Neil's lemma and (5)-(7). In particular it was Cianchi's  idea to exploit Adams' trace inequality (59) that eventually lead us to exploit instead Adams' weak-type estimates (21), in order to obtain a further substantial extension of O'Neil's lemma.\n\n\\medskip\nIn a third application, we consider Adams inequalities for sums of weighted Riesz potentials, i.e. for integral operators with kernel\n$$K(x,y)=\\sum_{j=1}^N g_j(x,y)|x+a_j-y|^{ d-n}$$\nwhere the functions $g_j$ are H\\\"older continuous, and where $x$ and $y$ are allowed to move in different domains. The sharp constant is explicitly described even in this case, see Theorem~15.\n\\smallskip\nFinally, and this was the original motivation for our work, we turn to the CR setting, by proving a sharpness result for some \nAdams' inequalities on the complex sphere, which were only partially proved  [BFM], using the methods of this work.\n\n\n\\medskip\n\nThe paper is organized in two main parts. In Part I we give the main results, in a measure-theoretical setting. Some portions of some  proofs are of course based on Adams' and O'Neil's original arguments,  but we decided to include them, in part because the  modifications are many, and often  not trivial,   and in part to achieve a rather self-contained and cleaner presentation. \n\n In Part II we give several new applications of the general results of Part I:  higher order Adams and Moser-Trudinger trace inequalities, Moser-Trudinger and  Adams inequalities for general and then specific  elliptic operators and parametrix-like potentials respectively, followed by those for sums of weighted Riesz potentials, and finally for certain types of potentials arising in CR geometry. \nSome  of these applications  could  be combined together, but we decided to keep them separate in order to highlight the relevant aspects of a given setting, rather than presenting more comprehensive theorems with too many parameters. \n\nWe certainly do not claim to have covered every possible Moser-Trudinger inequality, in fact we hope that many more could be  obtained using our setup, in a relatively painless way, and in a variety of settings. Moreover, in Theorems 10 and 12, a general form of the sharp exponential constant is given, but in specific cases it could be more helpful  to know this constant more explicitly. In this work we limit ourselves to give more explicit values in the case of  second order operators and certain other vector-valued inequalities, but more such computations are   possibile. \n\\eject\nAnother interesting situation arises regarding  Moser-Trudinger inequalities  in the space $W^{d,p}(\\Omega)$, i.e. without boundary conditions. In [Ci2] Cianchi obtained a sharp inequality for the case $W^{1,n}(\\Omega)$, but using different tools than ours, such as the isoperimetric inequality. It is possible that our methods are suitable to handle at least some special cases, such as low order operators, or particular domains.\n\n\n\\bigskip\\centerline{\\bf PART I: ABSTRACT THEOREMS}\\bigskip\n\n\n\n\n\\noindent {\\bf 1.  Adams inequalities on measure spaces}\\bigskip\n Let $(M,\\mu)$ be a measure space, and $\\mu$ a finite measure. \nGiven a  measurable $f:M\\to [-\\infty,\\infty]$ its distribution function will be denoted by \n$$m(f,s)=\\mu\\big(\\{x\\in M: |f(x)|>s\\}\\big),\\qquad s\\ge0$$\nits  nonincreasing rearrangement by\n$$f^*(t)=\\inf\\big\\{s\\ge0:\\,m(f,s)\\le t\\big\\},\\qquad t>0$$\nand \n$$f^{**}(t)={1\\over t}\\int_0^t f^*(s)ds,\\qquad t>0$$\nGiven another finite measure space $(N,\\nu)$ and  a $\\nu\\times\\mu-$measurable function $k:N\\times M\\to[-\\infty,\\infty]$ we let, for $t>0$, \n$$ k_1^*(t)=\\sup_{x\\in N} k^*(x,\\cdot)(t)$$\n$$k_2^*(t)=\\sup_{y\\in M} k^*(\\cdot,y)(t)$$\nwhere $k^*(x,\\cdot)(t)$ is the nonincreasing rearrangement of $k(x,y)$ with respect to the variable $y$ for fixed $x$, and $k^*(\\cdot,y)(t)$ is its analogue for fixed $y$. With a slight abuse of notation we set\n$$k_j^{**}(t)={1\\over t}\\int_0^t k_j^*(s)ds,\\qquad t>0,\\;j=1,2$$\nIf $k_2^*\\in L^1\\big([0,\\infty)\\big)$, or equivalently $m(k_2^*,\\cdot)\\in L^1\\big([0,\\infty)\\big)$, then the integral operator\n$$ Tf(x)=\\int_M k(x,y)f(y)d\\mu(y)\\eqno(12)$$\nis well defined and continuous from $L^1(M,\\mu)$ to $L^1(N,\\nu)$. In fact, as we shall see later, $Tf$ is also well defined on some $L^p$ under weaker integrability conditions on $k_2^*$, but with additional restrictions on $k_1^*$.\n\n\\medskip\nHere is our main theorem:\n\n\\proclaim Theorem 1. Let $k:N\\times M\\to[-\\infty,\\infty]$ be measurable on the finite measure space $(N\\times M,\\nu\\times\\mu)$ \nand  such that\n$$ m(k_1^*,s)\\le A s^{-\\beta}\\Big(1+O(\\log^{-\\gamma} s)\\Big)\\eqno(13)$$\n$$ m(k_2^*,s)\\le B s^{-\\beta_0}\\eqno(14)$$\nas $s\\to+\\infty$,  for some $\\beta,\\gamma>1$, $\\,0<\\beta_0\\le \\beta$ and $B>0$. Then, $T$ is  defined by (12) on $L^{\\beta'}(M)$ and \nthere exists a constant $C$ such that \n$$\\int_N \\exp\\bigg[{\\beta_0\\over  A\\beta}\\bigg({|Tf|\\over \\|f\\|_{\\beta'}}\\bigg)^\\beta\\,\\bigg]d\\nu\\le C\\eqno(15)$$\nfor each $f\\in L^{\\beta'}(M)$, with $\\displaystyle{1\\over \\beta}+{1\\over\\beta'}=1.$\n\\par\n\n\\noindent{\\bf Remarks.}\n\n\\noindent{\\bf 1.} \n It is possibile to modify slightly the arguments in order to include in Theorem 1 the case of Lorentz spaces. For simplicity we just treat $L^p$ spaces.\n\n\\medskip\n\\noindent{\\bf 2.} Theorem 1. holds verbatim in case $k$ is complex-valued and $T$ acts on complex-valued functions, provided that $|k(x,y)|$ satisfies conditions (13), (14).\\smallskip\n\nTheorem 1, as an immediate corollary of itself, \n  can be extended to vector-valued functions as follows. For a measurable $F:M\\to{\\hbox{\\mathbold\\char82}}^n$, $F=(F_1,...,F_n)$, define $|F|=(F_1^2+...+F_n^2)^{1\/2}$ and say $F\\in L^p(M)$ if $\\int_M|F|^p<\\infty$, likewise for vector-valued functions defined on $N$, valued on ${\\hbox{\\mathbold\\char82}}^n$, or  on $\\overline {\\hbox{\\mathbold\\char82}}^n=[-\\infty,\\infty]^n$. \n\n\n\\smallskip\n\\proclaim Theorem 1'.  Let  $K:N\\times M\\to\\overline{\\hbox{\\mathbold\\char82}}^n$, where $K=(K_1,...,K_n)$ be measurable and such that   $k(x,y)=|K(x,y)|$ satisfies conditions (13) and (14) of Theorem 1. If  \n$$TF(x)=\\int_M K(x,y)\\cdot F(y) \\,d\\mu(y)=\\int_M \\sum_{j=1}^n K_j(x,y)F_j(y)\\,d\\mu(y)$$\nthen, $T$ is defined on $L^{\\beta'}(M)$ and \nthere exists a constant $C$ such that \n$$\\int_N \\exp\\bigg[{\\beta_0\\over  A\\beta}\\bigg({|TF|\\over \\|F\\|_{\\beta'}}\\bigg)^\\beta\\,\\bigg]d\\nu\\le C\\eqno(16)$$\nfor each $F\\in L^{\\beta'}(M)$, with $\\displaystyle{1\\over \\beta}+{1\\over\\beta'}=1.$\n\\par\n\\eject\nThe formulation in terms of vector-valued function is useful since in many cases one has a  representation formula\nof a function which involves the  gradient operator, as in the classical Adams setting. Needless to say a similar version of the inequality holds for ${\\hbox{\\mathbold\\char67}}}\\def \\sC{{\\hbox{\\smathbold\\char67}}^n$-valued kernels and functions. It is important to point out that while the inequality of Theorem~1' is an immediate consequence of the scalar case, via Cauchy-Schwarz,  ths is not the case for the  sharpness statement  (see Theorem 4).\n \\medskip\n\nThe following elementary facts about rearragements will be useful ($f,g$ denote two measurable functions on $M$):\n\\smallskip\n\\noindent{\\bf Fact 1.}  $\\;m(f,s)=m(f^*,s)$ and $\\;m(f^*,s)\\le m(g^*,s)$ for all $s>s_0$ (some $s_0>0$) if and only if $f^*(t)\\le g^*(t)$ for all $t<t_0$ (some $t_0>0$).\\smallskip\n\\noindent{\\bf Fact 2.} If $\\psi(s)$ is continuous and strictly decreasing on $[s_0,\\infty)$ then \n$\\inf\\{s:\\,\\psi(s)\\le t\\}=\\psi^{-1}(t)$ for $t<\\psi(s_0)$, (and hence $\\psi$ is the distribution function of $\\psi^{-1}$ on that interval). \n\\smallskip\n\\noindent{\\bf Fact 3.} Given a measurable $k(x,y)$ on $N\\times M$, if $\\wtilde m(k,s)=\\sup_x m\\big(k^*(x,\\cdot),s\\big)=\\sup_x m\\big(k(x,\\cdot),s\\big)$ and $\\wtilde k(t)=\\inf\\big\\{s:\\, \\wtilde m(k,s)\\le t\\big\\}$, then $m(\\wtilde k,s)=\\wtilde m(k,s)$ and $\\wtilde k(t)=\\sup_x k^*(x,\\cdot)(t)$.\n\\smallskip\n\\noindent{\\bf Fact 4.} The following are equivalent ($A,\\beta,\\gamma>0$): \\smallskip\n\\item{a)} $\\;m(f^*,s)\\le As^{-\\beta}(1+C\\log^{-\\gamma}s),$ for all  $s>s_0>1$\n\\smallskip\\item{b)} $\\; f^*(t)\\le A^{1\/\\beta} t^{-1\/\\beta}(1+C'|\\log t|^{-\\gamma}),$ for all $t<t_0<1$. \n\\smallskip Likewise, the following are equivalent\\smallskip\n\\item{a')} $\\;m(f^*,s)\\ge As^{-\\beta}(1-C\\log^{-\\gamma}s)>0,$ for all $s>s_0>1$\n\\smallskip\\item{b')} $\\; f^*(t)\\ge A^{1\/\\beta} t^{-1\/\\beta}(1-C'|\\log t|^{-\\gamma})>0,$ for all $t<t_0<1$. \n\n\\def{\\rm supp}{{\\rm supp}}\n\n\\medskip\\medskip\nThe first, and crucial, step in the proof of Theorem 1 is the following $L^p$  version of O'Neil's lemma:\n\n\\medskip\n\\proclaim Lemma 2 (Improved O'Neil's Lemma). Let $k: N\\times M\\to [-\\infty,\\infty]$ be measurable and\n$$k_1^*(t)\\le M t^{-1\/\\beta},\\qquad k_2^*(t)\\le B t^{-1\/\\beta_0},\\qquad t>0\\eqno(17)$$\nwith $\\beta>1$ and $0<\\beta_0\\le \\beta$.\nIf  \n$$\\max\\Big\\{1,{\\beta-\\beta_0\\over\\beta-1}\\Big\\}< p<{\\beta\\over\\beta-1}=\\beta',\\qquad\\quad q={p\\beta_0\\over \\beta-(\\beta-1)p}>p\\eqno(18)$$ then $T$ is defined on $L^{\\beta'}(M)$, in fact $T:L^p(M)\\to L^{q,\\infty}(N)$ and bounded, and  there is a constant $C=C(M,B,\\beta,\\beta_0,p)$ such that for any $f\\in L^{\\beta'}(M)$\n$$ (Tf)^{**}(t)\\le C\\,\\max\\big\\{\\tau^{-{\\beta_0\\over  q \\beta}}, t^{-{1\\over q}}\\big\\}\\int_0^\\tau f^*(u) u^{-1+{1\\over p}}du+\\int_\\tau^\\infty f^*(u)k_1^*(u)du,\\quad \\forall t,\\tau>0.\\eqno(19)$$\nIf instead of (17) we assume $\\,k_1^*, k_2^*\\in L^1\\big([0,\\infty)\\big)$ then for every $f\\in L^1(M)$\n$$(Tf)^{**}(t)\\le \\tau\\, \\max\\big\\{k_1^{**}(\\tau),k_2^{**}(t)\\big\\}\\, f^{**}(\\tau)+\\int_\\tau^\\infty f^*(u)k_1^*(u)du,\\quad \\forall t,\\tau>0.\\eqno(20)$$\n\\par \\smallskip\n We observe that  inequality (20) implies (19) in case $\\beta_0>1$, that is when both    \n $k_1^*$ and $k_2^*$ are integrable, and it is also perfectly suitable to prove Theorem 1 in that case, but it is useless when $\\beta_0\\le 1$.\n\\medskip\n\\pf Proof.  We begin right away with the following weak-type estimate due to Adams [Ad3]. If $k$ and $f$ are nonnegative,  with \n$$\\sup_{x\\in N} \\,m\\big(k(x,\\cdot),s)\\le M s^{-\\beta},\\qquad \\sup_{y\\in M} \\,m\\big(k(\\cdot,y),s\\big)\\le Bs^{-\\beta_0}$$\n which are equivalent to (17), and under the hypothesis  (18), then for $s>0$\n$$ s\\, m(Tf,s)^{1\\over q}=s\\,\\nu\\big(\\{x: Tf(x)>s\\}\\big)^{1\\over q}\\le {q^2\\over \\beta_0(q-p)}M^{1-{1\\over p}}B^{1\\over q} \\|f\\|_p\\eqno(21)$$\nor\n$$ (Tf)^*(t)\\le C t^{-{1\\over q}} \\|f\\|_p,\\qquad \\forall t>0.\\eqno(22)$$\nThis means that $T:L^p(M)\\to L^{q,\\infty}(N)$ is bounded, in particular  $T$ is well defined on $L^{\\beta'}(M)\\subseteq L^p(M)$. \n\nWithout loss of generality we can assume throughout this proof that both $k$ and $f$ are nonnegative. With a slight abuse of language we let ${\\rm supp}(f)=\\{x\\in M:\\,f(x)\\neq0\\}$. The main step of the proof relies on the following:\n\\medskip\n\\proclaim Claim (See also Lemma 1.4 in [ON]). If $\\mu({\\rm supp} f)=z$ and $0\\le f(z)\\le \\alpha$, and if $k_1^*,k_2^*$ satisfy conditions (17), then $\\forall t>0$\n$$(Tf)^{**}(t)\\le\\alpha \\,z\\,k_1^{**}(z).\\eqno(23)$$\n$$(Tf)^{**}(t)\\le C\\, \\alpha\\, z^{{1\\over p}} t^{-{1\\over q}}.\\eqno(24)$$\nIf instead of $(17)$ we assume that $k_1^*$ and $k_2^*$ are integrable, then (23) holds and (24) can be replaced by \n$$(Tf)^{**}(t)\\le\\alpha \\,z\\,k_2^{**}(t)\\eqno(25).$$\\par\\medskip\nAssuming the Claim, the proof of the Lemma proceeds as follows.\nFor fixed $t,\\tau>0$, pick  $\\{y_n\\}_{-\\infty}^\\infty$ such that $y_0=f^*(\\tau),\\,y_n\\le y_{n+1}, \\,y_n\\to+\\infty$\nas $n\\to+\\infty$, and $y_n\\to0$ as $n\\to-\\infty$.  Then \n$$f(y)=\\sum_{-\\infty}^\\infty f_n(y)\\quad{\\hbox{where}}\\quad f_n(y)=\\cases{0 &if $\\;f(y)\\le y_{n-1}$\\cr f(y)-y_{n-1} \n&if $\\;y_{n-1}<f(y)\\le y_n$ \\cr y_n-y_{n-1} & if $\\;y_n<f(y).$\\cr}$$\n\nObserve that supp$f_n\\subseteq E_n:=\\big\\{y: f(y)>y_{n-1}\\big\\}$,  $\\;\\mu(E_n)=m(f,y_{n-1})$, and also  $\\;0\\le f_n(y)\\le y_n-y_{n-1}$. Write \n$$f=\\sum_{-\\infty}^0f_n+\\sum_1^{\\infty}f_n=g_1+g_2$$\nso that  $(Tf)^{**}\\le (Tg_1)^{**}+(Tg_2)^{**}$ (this is the subadditivity of $(\\cdot)^{**}$). Using the Claim (24) we obtain\n$$(Tg_2)^{**}(t)\\le \\sum_1^\\infty (Tf_n)^{**}(t)\\le Ct^{-{1\\over q}}\\,\\sum_1^{\\infty}(y_n-y_{n-1})\\big(m(f,y_{n-1})\\big)^{{1\\over p}}$$\nso that taking the inf over all such $\\{y_n\\}$ we get\n$$\\eqalign{&(Tg_2)^{**}(t)\\le Ct^{-{1\\over q}}\\int_{f^*(\\tau)}^\\infty \\big(m(f,y)\\big)^{{1\\over p}} dy=-\\int_0^\\tau\\big(m(f,f^*(u))\\big)^{{1\\over p}}d\\,f^*(u)\\cr&\\le- \\int_0^\\tau u^{{1\\over p}} d\\,f^*(u)=-u^{{1\\over p}}f^*(u)\\Big|_0^\\tau+{1\\over p}\\int_0^\\tau u^{-1+{1\\over p}}f^*(u)du\\le{1\\over p}\\int_0^\\tau u^{-1+{1\\over p}}f^*(u)du. \\cr}  $$\n(the last inequality follows since $f\\in L^{\\beta'}\\Longrightarrow t^{{1\\over \\beta'}}f^*(t)\\to0$, as $t\\to0$.)\n\nLikewise, using the Claim (23)\n$$(Tg_1)^{**}(t)\\le \\sum_{-\\infty}^0 (Tf_n)^{**}(t)\\le \\sum_{-\\infty}^0 (y_n-y_{n-1})m(f,y_{n-1})k_1^{**}(m(f(y_{n-1}))$$\nand so\n$$\\eqalign{&(Tg_1)^{**}(t)\\le\\int_0^{f^*(\\tau)} m(f,y)k_1^{**}\\big(m(f,y)\\big)dy=\n-\\int_\\tau^\\infty m\\big(f,f^*(u)\\big)k_1^{**}\\Big(m\\big(f,f^*(u)\\big)\\Big) df^*(u)\\cr&=-\\int_\\tau^\\infty u \\,k_1^{**}(u)df^*(u)=\n-u\\, k_1^{**}(u)f^*(u)\\bigg|_\\tau^\\infty+\\int_\\tau^\\infty k_1^*(u)f^*(u)du\\cr&\\le \\tau\\,k_1^{**}(\\tau)f^*(\\tau)+\\int_\\tau^\\infty f^*(u)k_1^*(u)du\\le \\tau^{1-{1\\over p}}\\,k_1^{**}(\\tau)\\int_0^\\tau f^*(u)u^{-1+{1\\over p}}du+\\int_\\tau^\\infty f^*(u)k_1^*(u)du\\cr&\n\\le C\\,\\tau^{1-{1\\over p}-{1\\over \\beta}}\\int_0^\\tau f^*(u)u^{-1+{1\\over p}}du+\\int_\\tau^\\infty f^*(u)k_1^*(u)du\n}$$\nand (19) follows since $\\displaystyle{{1\\over p}+{1\\over \\beta}-1={\\beta_0\\over q\\beta}}$.\n\n To prove (20), assume that $k_1^*,\\,k_2^*$ and $f$ are integrable and estimate $(Tg_1)^{**}$ as before. The estimate for $(Tg_2)^{**}$ is now performed as above, but using (25) instead of (24). This yields\n\n$$\\eqalign{(Tf)^{**}(t)\\le  \\max\\big\\{k_1^{**}(\\tau),k_2^{**}(t)\\big\\}\\,\\bigg[\\tau f^*(\\tau )&+\\int_{f^*(\\tau )}^\\infty m(f,y)dy\\bigg]+\\int_\\tau ^\\infty f^*(u)k_1^*(u)du\\cr}$$\nand (20) follows from the identity\n$$ \\int_{f^*(\\tau )}^\\infty m(f,y)dy=\\int_{f^*(\\tau )}^\\infty m(f^*,y)dy=\\int_0^\\tau  f^*(u)du-\\tau  f^*(\\tau ).$$\n\n\n\\medskip\n\\smallskip\\noindent{\\bf Proof of Claim.}  Let $r>0$ and   set \n$$k_r(x,y)=\\cases{k(x,y) & if $\\;k(x,y)\\le r$\\cr\\cr r & otherwise,\\cr}\\qquad k(x,y)=k_r(x,y)+k^r(x,y).$$\nso that \n$$Tf(x)=\\int_M k_r(x,y)f(y)d\\mu(y)+\\int_M k^r(x,y )f(y)d\\mu(y)=h_1(x)+h_2(x).$$\nAssume that $k_1^*$ is integrable. Then,\nfor every given $x$\n$$ h_2(x)\\le \\|f\\|_\\infty^{}\\int_M k^r(x,y)d\\mu(y)\\le \\alpha \\int_r^\\infty m(k_1^*,s)ds,\\eqno(26)$$\n$$ h_1(x)\\le \\|f\\|_1^{}\\sup_y k_r(x,y)\\le \\alpha z r,\\eqno(27)$$\nso that letting $r=k_1^*(z)$ in (26) and (27) leads to\n$$\\eqalign{(Tf)^{**}(t)&={1\\over t}\\int_0^t (Tf)^*\\le \\|Tf\\|_\\infty^{}\\le \\|h_1\\|_\\infty^{}+ \\|h_2\\|_\\infty^{}\\cr&\\le\n\\alpha z \\,k_1^*(z)+\\alpha\\int_{k_1^*(z)}^\\infty m(k_1^*,s)ds=\\alpha\\int_0^z k_1^*(s)ds=\\alpha z \\,k_1^{**}(z),\\cr}$$\nwhich is (23). \nIf in addition $k_2^*$ is integrable, then \n$$\\eqalign{&\\int_N h_2(x)d\\nu(x)=\\int_N d\\nu(x)\\int_M k^r(x,y)f(y)d\\mu(y)\\cr&=\\int_M f(y)\\bigg(\\int_N k^r(x,y)d\\nu(x)\\bigg)d\\mu(y)\n\\le \\|f\\|_1^{}\\int_r^\\infty m(k_2^*,s)ds\\le \\alpha z \\int_r^\\infty m(k_2^*,s)ds,\\cr}\\eqno(28)$$\ntherefore,   letting $r=k_2^*(t)$ and using (27) and (28)\n$$\\eqalign{t\\,(Tf)^{**}(t)&\\le \\int_0^t h_1^*+\\int_0^t h_2^* \\le t\\,\\|h_1\\|_\\infty^{} +\\int_0^\\infty h_2^*\\cr& \\le t\\,\\alpha z\\, k_2^*(t)+\\alpha z\\int_{k_2^*(t)}^\\infty m(k_2^*,s)ds=\\alpha z\\,t\\, k_2^{**}(t)\\cr}$$\nand this concludes the proof of  (23) and (25), in case both $k_1^*$ and $k_2^*$ are integrable.\nIf conditions (17) are assumed instead, then (23) still holds (since only integrability of $k_1^*$ was needed), and  \nestimate (24) is an immediate consequence of the weak-type estimate (22).\n $$\\eqno\/\\!\/\\!\/$$\n\n\n\\bigskip\\noindent{\\bf Remark.} We emphasize here the new elements appearing in the Lemma, as compared to O'Neil's original version. First, the  role of the two measures, as reflected in the explicit dependence on $k_1^*$ and $k_2^*$, and their bounds. Secondly,  the fact that O'Neil's lemma is really a two-variable statement; this is hinted in the Claim, even in O'Neil's original version,  but it does not seem to have been noticed before. Our original version of the Lemma was just (20) with  $\\tau=t$ which was suitable to prove Theorem 1 when $\\beta_0=\\beta$ (our first version) but not for $\\beta_0<\\beta$. The further improvements of O'Neil's lemma came about in our attempts to incorporate some of Cianchi's main results [Ci1] in our general framework (see Theorem 6).\n\n\n\n\\vskip1em\n\n\n\\pf Proof of Theorem 1. It is enough to assume that $k$ is  nonnegative, and show that for each nonnegative $f\\in L^{\\beta'}(M)$    with $\\,\\|f\\|_{\\beta'}^{}\\le 1$ we have \n$$\\int_N \\exp\\bigg[{\\beta_0\\over A\\beta}(Tf)^\\beta\\,\\bigg]d\\nu\\le C\\eqno(29)$$\nfor some $C$ independent of $f$.\n\n\nPick any $p$ as in (18). By (19) of the improved   O'Neil's Lemma 2, with $\\tau=t^{\\beta\/\\beta_0}$ \n$$\\eqalign{&(Tf)^*(t)\\le(Tf)^{**}(t)\\le C t^{-{1\\over q}}\\int_0^{t^{\\beta\/\\beta_0}}f^*(u) u^{-1+{1\\over p}}du+\\int_{t^{\\beta\/\\beta_0}}^\\infty k_1^*(u) f^*(u)du\\cr &=\nC t^{-{1\\over q}}\\int_0^{t}f^*\\big(u^{\\beta\\over\\beta_0}\\big) u^{-1+{\\beta\\over p\\beta_0}}du+{\\beta\\over\\beta_0}\\int_t^\\infty k_1^*\\big(u^{\\beta\\over\\beta_0}\\big) f^*\\big(u^{\\beta\\over\\beta_0}\\big)u^{{\\beta\\over\\beta_0}-1}du.\\cr}\\eqno(30)$$\nBy Fact 4, combined with the fact that $k_j^*(t)=0$ for $t\\ge\\max\\{\\nu(N),\\mu(M)\\}$,\n$$k_1^*\\big(u^{\\beta\\over\\beta_0}\\big) \\le A^{1\\over\\beta}u^{-{1\\over\\beta_0}}\\big(1+C(1+|\\log u|)^{-\\gamma}\\big)\\,,\\quad u>0\\eqno(31)$$\n(C denotes a positive constant that may change from place to place).\n\nCombining (30) and (31) yields\n$$ \\eqalign{(Tf)^{**}(t)\\le C & t^{-{1\\over q}}\\int_0^{t}f^*\\big(u^{\\beta\\over\\beta_0}\\big) u^{-1+{\\beta\\over p\\beta_0}}du\n+\\cr&+{\\beta\\over\\beta_0}\\int_t^{\\mu(M)^{\\beta_0\/\\beta}}\\!\\!A^{1\/\\beta}\\big(1+C(1+|\\log u|)^{-\\gamma}\\big)f^*\\big(u^{\\beta\\over\\beta_0}\\big)u^{{\\beta\\over\\beta_0}-1}du\\cr}\n$$\nand therefore, with $t_1=\\max\\{\\nu(N),\\mu(M)^{\\beta_0\/\\beta}\\}$,\n$$\\eqalign{&\\int_N \\exp\\bigg[{\\beta_0\\over A\\beta}(Tf)^\\beta\\,\\bigg]d\\nu(x)=\\int_0^{\\nu(N)} \\exp\\bigg[{\\beta_0\\over A\\beta}\\big((Tf)^*(t)\\big)^\\beta\\bigg]dt\n\\le\\int_0^{\\nu(N)}\\exp\\bigg[{\\beta_0\\over A\\beta}\\Big((Tf)^{**}(t)\\Big)^\\beta\\Big]dt\\cr&\\le\\int_0^{t_1} \\exp\\bigg[\\bigg( C t^{-{1\\over q}}\\int_0^{t}f^*\\big(u^{\\beta\\over\\beta_0}\\big) u^{-1+{\\beta\\over p\\beta_0}}du+\\cr&\\hskip3em      \n+ \\bigg({\\beta\\over\\beta_0}\\bigg)^{1\\over\\beta'}\\int_t^{t_1}\\big(1+C(1+|\\log u|)^{-\\gamma}\\big)f^*\\big(u^{\\beta\\over\\beta_0}\\big)u^{{\\beta\\over\\beta_0}-1}du\\bigg)^\\beta\\,\\bigg]dt.\\cr}$$\n\nNow we make the changes of variables $u=e^{-x},\\, t=e^{-y}$, and we let $y_1=-\\log t_1$ and \n$$\\phi(x)=\\bigg({\\beta\\over\\beta_0}\\bigg)^{1\\over\\beta'}f^*\\big(e^{-{\\beta x\\over\\beta_0}}\\big)\\,e^{-{\\beta-1\\over\\beta_0}x}.$$ Notice  that $\\phi$ is defined  on $[y_1,\\infty)$ and $\\|\\phi\\|_{\\beta'}^{}=\\|f^*\\|_{\\beta'}^{}=\\|f\\|_{\\beta'}^{}\\le1$.\n\nWith these changes, estimate (29) reduces to\n$$\\int_{y_1}^\\infty \\exp\\bigg[\\bigg( H\\int_{y}^\\infty \\phi(x) e^{y-x\\over q}dx+\\int_{y_1}^y\\Big(1+H(1+|x|)^{-\\gamma}\\Big)\\phi(x)dx\\bigg)^\\beta-y\\bigg]\\,dy\\le C\\eqno(32)$$\nwhere $H$ is a suitable, fixed, positive constant. \n\nDefine \n$$g(x,y)=\\cases{1+H(1+|x|)^{-\\gamma} & if $y_1\\le x\\le y$ \\cr \\cr\n H e^{{y-x\\over q}} & if $y_1\\le y<x<\\infty$ \\cr}\\eqno(33)$$\nand  \n$$F(y)=y-\\bigg(\\int_{y_1}^\\infty g(x,y)\\phi(x) dx\\bigg)^\\beta\\eqno(34)$$\nwhich is defined for $y\\in[y_1,\\infty)$. Estimate (32) is a direct consequence of the following modified Adams-Garsia's lemma:\n\n\\bigskip\n\\proclaim Lemma 3. Suppose that $\\phi:[y_1,\\infty)\\to [0,\\infty)$ satisfies $\\displaystyle{\\int_{y_1}^\\infty \\phi^{\\beta'}\\le 1},$ and  $g$ and $F$ be defined as in (33), (34), with $H>0$, $\\beta>1$, $q>0$ \n and $\\displaystyle{{1\\over\\beta}+{1\\over\\beta'}=1}$. Then there exists \na constant $C$ independent   of $\\phi$ such that \n$$\\int_{y_1}^\\infty e^{-F(y)}dy\\le C.\\eqno(35)$$\n\\par\n\nThis lemma  differs from the original Adams-Garsia\n lemma (Lemma 1 in [A]) by the perturbation term $H(1+|x|)^{-\\gamma}$ for $x\\le y$ (which  was not present in  Adams-Garsia's lemma). In his original work Moser had 1 for $x\\le y$ and 0 for $x>y$ which makes the argument much simpler. The proof below is a modification of the proof or Lemma 3.2 in [F], which was itself a modification of the proof of Lemma 1 in [A]. We note that in [FFV] there is an even more general version of Lemma 3, which appeared after that in [F], but we decided to include its  proof in order to make our results  self contained.\n\n\\bigskip\n\\pf Proof of Lemma 3. Let $E_\\lambda=\\{y\\ge y_1:\\, F(y)\\le \\lambda\\}$ and let $|E_\\lambda|$ be its Lebesgue measure.  Then\n$$\\int_{y_1}^\\infty e^{-F(y)}dy=\\int_{-\\infty}^\\infty |E_\\lambda| e^{-\\lambda}d\\lambda.$$\n\n\n\\proclaim Claim 1. There exists $c\\ge0$ independent of $\\phi$ such that if $E_\\lambda\\neq \\emptyset$, then $\\lambda\\ge -c$, i.e.\n$\\inf_{y\\ge y_1} F(y)\\ge -c>-\\infty$.\\par \\medskip\n\\proclaim Claim 2. There exist $C$ independent of $\\phi$ and $\\lambda$ such that for every $\\lambda\\in {\\hbox{\\mathbold\\char82}}$\n\n$$|E_\\lambda|\\le C(1+|\\lambda|).\\eqno(36)$$\n\\par\n\nClaims 1 and 2 imply (35) since \n$$\\int_{y_1}^\\infty e^{-F(y)}dy=\\int_{-c}^\\infty |E_\\lambda| e^{-\\lambda}d\\lambda\\le C\\int_{-c}^\\infty (1+|\\lambda|)e^{-\\lambda}d\\lambda,$$\nwhich is a constant independent of $\\phi$.\n\n\\medskip\n\\pf Proof of Claim 1.  It is enough to assume that $\\lambda<0$ and $y_1-\\lambda>0$. If $y\\in E_\\lambda$ then  \n$$\\eqalign{&(y-\\lambda)^{1\\over\\beta}\\le\\int_{y_1}^y\\Big(1+H(1+|x|)^{-\\gamma}\\Big)\\phi(x)dx+H\\int_y^\\infty e^{y-x\\over q}\\phi(x)dx\\cr\n& \\le \\bigg(\\int_{y_1}^y \\phi^{\\beta'}\\bigg)^{1\\over\\beta'}\\bigg(\\int_{y_1}^y \\Big(1+H(1+|x|)^{-\\gamma}\\Big)^\\beta dx\\bigg)^{{1\\over\\beta}}+H\n\\bigg(\\int_y^\\infty\\phi^{\\beta'}\\bigg)^{1\\over\\beta'}\\bigg(\\int_y^\\infty e^{(y-x){\\beta\\over q}}dx\\bigg)^{1\\over\\beta}.\\cr}$$\nNote that for $a,b\\ge0$ and $\\beta\\ge1$ \n$$(a+b)^\\beta\\le a^{\\beta}+\\beta 2^{\\beta-1}(a^{\\beta-1}b+b^{\\beta})\\eqno(37)$$\n(identity at $b=0$, and $b-$derivative of LHS smaller than $b-$derivative of RHS). Hence\n$$\\int_{y_1}^y \\Big(1+H(1+|x|)^{-\\gamma}\\Big)^\\beta dx\\le\\int_{y_1}^y \\Big(1+H_1(1+|x|)^{-\\gamma}\\Big) dx\\le y-y_1+d_1= y+d$$\nsome $d\\in{\\hbox{\\mathbold\\char82}}$, independent of $y$ (here is where we use $\\gamma>1$).\n\nAs a result, if we let \n$$L(y)=\\bigg(\\int_y^\\infty \\phi^{\\beta'}\\bigg)^{1\\over\\beta'}\\in [0,1].$$\n we have (using (37) again)\n$$\\eqalign{y-\\lambda&\\le \\Big[\\big(1-L(y)^{\\beta'}\\big)^{{1\\over\\beta'}}(y+d)^{{1\\over\\beta}}+CL(y)\\Big]^\\beta\\cr&\\le\n\\big(1-L(y)^{\\beta'}\\big)^{{\\beta\\over\\beta'}}(y+d)+\\beta 2^{\\beta-1}\\Big[\\big(1-L(y)^{\\beta'}\\big)^{\\beta-1\\over\\beta'}(y+d)^{\\beta-1\\over\\beta} CL(y)+C^\\beta L(y)^\\beta\\Big]\\cr}$$\n\nSince $\\beta,\\beta'>1, \\,L(y)\\in[0,1]$ and   $\\big(1-L(y)^{\\beta'}\\big)^{\\beta\\over\\beta'}\\le 1-{\\displaystyle{1\\over\\beta'}}L(y)^{\\beta'}$,\n if we let   $z=(y+d)^{1\/\\beta'}L(y)\\ge0$ we obtain\n$$z^{\\beta'}\\le D z+\\beta' \\lambda+D$$\nfor some constant $D$ (independent of $y$ and $\\phi$.\nSince $z^{\\beta'}-Dz-D$ has a finite negative minimum on $[0,\\infty)$, we deduce that if $E_\\lambda\\neq\\emptyset$ then \n$\\lambda\\ge -c$, for some $c\\ge0$ (independent of $y$ and $\\phi$). \n\n Note also that for large $z$ we have $Dz\\le {1\\over2} z^{\\beta'}$ so that  $z^{\\beta'}\\le C(|\\lambda|+1)$\nor\n$$ (y+d)^{{1\\over\\beta'}}L(y)\\le C(|\\lambda|^{{1\\over\\beta'}}+1)\\eqno(38)$$\nfor some $C$ independent of $y$, $\\phi$, and $\\lambda$.\n\\bigskip\n\\pf Proof of Claim 2. \nIt is enough to  prove that there exist $H>0$ (independent of $\\phi$) such that for any $\\lambda\\in{\\hbox{\\mathbold\\char82}}$\n$$t_1,t_2\\in E_\\lambda\\, {\\hbox { and }}\\, t_2>t_1>H|\\lambda|+H\\,\\Longrightarrow \\, t_2-t_1\\le H|\\lambda|+H.\\eqno(39)$$\nIndeed, if this is the case, then (recall that $E_\\lambda\\subseteq[y_1,\\infty)$)\n$$\\eqalign{|E_\\lambda|&=\\big|E_\\lambda\\cap\\{t: t\\le H|\\lambda|+H\\}\\big|+\\big|E_\\lambda\\cap\\{t:t>H|\\lambda|+H\\}\\big|\\cr& \\le H|\\lambda|+H-y_1+\\sup_{t_2>t_1>H|\\lambda|+H\\atop t_1,t_2\\in E_\\lambda} (t_2-t_1)\\le C|\\lambda|+C.\\cr} $$\n\nTo show (39), pick $t_1,t_2\\in E_\\lambda$, $\\;t_2>t_1$, so that, arguing  as in the proof of Claim 1\n$$\\eqalign{(t_2-\\lambda)^{{1\\over\\beta}}&\\le \\int_{y_1}^\\infty g(x,t_2)\\phi(x)dx\\le \\bigg(\\int_{y_1}^{t_1}g(x,t_2)^\\beta\\bigg)^{{1\\over\\beta}}\\bigg(\\int_{y_1}^{t_1}\\phi^{\\beta'}\\bigg)^{{1\\over\\beta'}}\\cr& \\hskip2em + \\bigg(\\int_{t_1}^{t_2}g(x,t_2)^\\beta\\bigg)^{{1\\over\\beta}}\\bigg(\\int_{t_1}^{t_2}\\phi^{\\beta'}\\bigg)^{{1\\over\\beta'}}+\\bigg(\\int_{t_2}^{\\infty}g(x,t_2)^\\beta\\bigg)^{1\\over\\beta}\\bigg(\\int_{t_2}^{\\infty}\\phi^{\\beta'}\\bigg)^{{1\\over\\beta'}}\\cr&\n\\le (t_1+d)^{{1\\over\\beta}}+(t_2-t_1+d_1)^{{1\\over\\beta}}\\bigg(\\int_{t_1}^\\infty\\phi^{\\beta'}\\bigg)^{{1\\over\\beta'}}+C\\bigg(\\int_{t_1}^\\infty\\phi^{\\beta'}\\bigg)^{{1\\over\\beta'}}\\cr &=(t_1+d)^{{1\\over\\beta}}+\\big((t_2-t_1+d)^{{1\\over\\beta}}+C\\big) L(t_1)\\cr}$$\nwhich , using (37) and (38), implies\n$$\\eqalign{t_2-\\lambda&\\le t_1+d+\\beta 2^{\\beta-1}\\bigg[(t_1+d)^{\\beta-1\\over\\beta}\\big((t_2-t_1)^{{1\\over\\beta}}+C\\big)L(t_1)+\\big((t_2-t_1)^{{1\\over\\beta}}+C\\big)^\\beta L(t_1)^\\beta\\bigg]\\cr &\\le\n t_1+d+\\beta 2^{\\beta-1}\\bigg[\\big((t_2-t_1)^{{1\\over\\beta}}+C\\big)(t_1+d)^{{1\\over\\beta'}}L(t_1)+2^\\beta(t_2-t_1)L(t_1)^\\beta+2^\\beta C^\\beta\\bigg]\\cr&\\le\n t_1+\\big((t_2-t_1)^{{1\\over\\beta}}+C\\big)(C|\\lambda|^{{1\\over\\beta'}}+C)+C(t_2-t_1)L(t_1)^\\beta+C\\cr&\n\\le t_1+{t_2-t_1\\over\\beta}+ {(C|\\lambda|^{{1\\over\\beta'}}+C)^{\\beta'}\\over\\beta'}+C(t_2-t_1)L(t_1)^\\beta+C|\\lambda|^{{1\\over\\beta'}}+C.\\cr}$$\n\nHence, \n$$ {t_2-t_1\\over\\beta'}\\le C|\\lambda|+C+C(t_2-t_1)L(t_1)^\\beta\\le C|\\lambda|+C+(t_2-t_1){C|\\lambda|+C\\over t_1+d}$$\nso it follows that there is $C$ so that \n$$t_1+d>2\\beta' C|\\lambda|+2\\beta' C\\,\\Longrightarrow \\, t_2-t_1\\le 2\\beta'C|\\lambda|+2\\beta' C,$$ which is (39). \nClaim 2, Lemma 3 and Theorem 1 are thus completely proven.$$\\eqno\/\\!\/\\!\/$$\n\n\\bigskip\\medskip\n\\noindent{\\bf 2. Conditions for sharpness}\n\\bigskip\nIn the following theorem we prove that,   under suitable ``geometric\" conditions, equality in (13),  implies that $\\displaystyle{\\beta_0\\over A\\beta}$\nin (15) or (16) is sharp, i.e. it cannot be replaced by a larger constant. We state and prove the general vector-valued case, since it does not follow directly from the scalar case, as opposed to the proof of Theorem 1'. It will be apparent from the proof that the same result will also hold for complex-valued operators (see Remark 1 after the proof of Theorem 4).\n\\smallskip\nFor measurable $F:M\\to{\\hbox{\\mathbold\\char82}}^n$ and  $K:N\\times M\\to\\overline{\\hbox{\\mathbold\\char82}}^n$ let  \n$$TF(y)=\\int_M K(x,y)\\cdot F(y)\\,d\\mu(y)$$\nif the integral is well defined.\n\n\\bigskip\\eject\n\n\\proclaim Theorem 4. Suppose that $k(x,y)=|K(x,y)|$ satisfies\n$$ m(k_1^*,s)=A s^{-\\beta}\\big(1+O(\\log^{-\\gamma} s)\\big),\\qquad {\\hbox { as }} s\\to+\\infty,\\eqno(40)$$\nor equivalently \n$$ k_1^*(t)=A^{1\/\\beta} t^{-1\/\\beta} \\Big(1+O\\big(|\\log t|^{-\\gamma}\\big)\\Big),\\qquad {\\hbox { as }} t\\to0,\\eqno(41)$$\n and \n$$ m(k_2^*,s)\\le B s^{-\\beta_0}$$\nas $s\\to+\\infty$,  for some $\\beta,\\gamma>1$, $\\,0<\\beta_0\\le \\beta$ and $B>0$.\nSuppose that there exist $x_m\\in N$, measurable sets $B_m\\subseteq N,\\, E_m\\subseteq M$, $\\,m\\in {\\hbox{\\mathbold\\char78}}}\\def\\S{{\\cal S} $, with the following properties:\\medskip\n\\noindent {a)} $E_m\\supseteq\\{y:\\,|K(x_m,y)|>m\\},$  $\\;\\mu(E_m)=O(m^{-\\beta})$, as $m\\to \\infty$\\smallskip\n\\noindent{b)} there exist constants $c_1,c_2>0$ such that $c_1m^{-\\beta_0}\\le \\nu(B_m)\\le  c_2m^{-\\beta_0},\\, m=1,2....$\n\\smallskip\n\\noindent{c)} \n$$k^*(x_m,\\cdot)(t)\\ge A^{1\/\\beta}t^{-1\/\\beta}\\Big(1-c_3|\\log t|^{-\\gamma}\\Big),\\quad 0<t<t_0<1\\eqno(42)$$\n\\noindent{d)}\n$$ \\int_{M\\setminus E_m} \\big|\\big(K(x,y)- K(x_m,y)\\big)\\cdot K(x_m,y)\\big|\\, |K(x_m,y)|^{\\beta-2} d\\mu(y)\\le c_4\\,,\\qquad \\forall x\\in B_m\n\\eqno(43)$$\nwith $c_3,c_4$ independent of $x$ and $m$. Then, the Adams inequality (16) holds and it is sharp, in the sense that \n$$\\sup_{F\\in L^{\\beta'}(M)} \\int_N \\exp \\bigg[\\alpha\\,\\bigg({|TF|\\over\\|F\\|_{\\beta'}^{}}\\bigg)^\\beta\\,\\bigg]d\\nu=+\\infty,\\qquad \\forall \\alpha>{\\beta_0\\over A\\beta}.$$\nMore specifically, if a), b) , c) hold and \n$$\\Phi_m(y)=K(x_m,y)|K(x_m,y)|^{\\beta-2}\\chi_{M\\setminus E_m}^{}(y)\\eqno(44)$$ then $\\Phi_m\\in\n L^{\\beta'}$ with \n$$\\|\\Phi_m\\|_{\\beta'}^{\\beta'}=A\\,\\log{1\\over\\mu(E_m)}+O(1),\\eqno(45)$$\nand if d) also holds then\n$$\\lim_{m\\to\\infty}\\int_N \\exp \\bigg[\\alpha\\,\\bigg({|T\\Phi_m|\\over\\|\\Phi_m\\|_{\\beta'}^{}}\\bigg)^\\beta\\,\\bigg]d\\nu=+\\infty,\n\\qquad \\forall \\alpha>{\\beta_0\\over A\\beta}.\\eqno(46) $$\n\\par\n\\def{\\rm sgn}{{\\rm sgn}}\n\\bigskip\\eject\n\\noindent{\\bf Remarks.} \\smallskip\n\n\\noindent{\\bf 1.} If there is a point $x_0$ such that $k_1^*(t)=k^*(x_0,\\cdot)(t)$ for small $t$, then typically one can choose $x_m=x_0$, so that (42) is automatically true. In the context of metric spaces one can typically choose $E_m$ to be the $m-$th level set of $|K(x_0,y)|$, or a possibly slightly larger set, and $B_m$  a suitable small ball around $x_0$.  In all the applications we know, the only minor technical check  is about  the integral estimate in (43), which  is usually a consequence of H\\\"older continuity  estimates on $K(x,y)$. This point is illustrated clearly in all the applications presented in  section 5.\n\n\\smallskip\n\\noindent{\\bf 2.} In the scalar case  $K(x,y)=k(x,y)$ condition d) obviously becomes\n$$\\int_{M\\setminus E_m} |k(x,y)- k(x_m,y)|\\, |k(x_m,y)|^{\\beta-1} d\\mu(y)\\le c_4\\,,\\qquad \\forall x\\in B_m.\\eqno(47)$$\nIn the vector-valued case condition d) is implied by \n$$\\int_{M\\setminus E_m}|K(x,y)- K(x_m,y)|\\, |K(x_m,y)|^{\\beta-1} d\\mu(y)\\le c_4\\,,\\qquad \\forall x\\in B_m.$$\n\\smallskip\n\\noindent{\\bf 3.} The classical form of a  Moser-Trudinger inequality for a differential (or pseudodifferential) operator of order $d$ takes the form\n$$\\int_N \\exp \\bigg[\\alpha\\,\\bigg({|u|\\over\\|Pu\\|_{p}^{}}\\bigg)^{p'}\\,\\bigg]d\\nu\\le C\\eqno(48)$$\nwhere $P$  acts on a suitable subspace of $L^p(N)$ (usually a Sobolev space). A  lower bound for $\\alpha$ can be achieved via a \n representation formula $u=T(Pu)$, where $T$ is an integral operator with kernel $K$, satisfying the hypothesis of Theorem 1 or 1'. If  the conditions in Theorem 4 are satisfied, then the sharpness of the constant follows immediately if one is able to produce a sequence $u_m$ in the given space such that $Pu_m=\\Phi_m$, the extremizing sequence of Theorem 4. When dealing with scalar functions this is usually possible (see theorems 6 and 10). Another similar  way to obtain an upper bound for $\\alpha$ is  by choosing a suitable sequence of functions $u_m$ and sets $B_m\\subseteq N$ such that $u_m\\ge \\delta_m$ on $B_m$, via the inequality \n$$\\alpha\\le \\liminf_n \\bigg({\\|Pu_m\\|_p\\over\\delta_m}\\bigg)^{p'}\\log{1\\over \\nu(B_m)}\\eqno(49)$$\nwhich follows easily from (48). This approach is slightly more flexible in that the  $u_m$ may not be the exact inverse images of the $\\Phi_m$,  even though they usually differ from those by  negligible terms. \n\n\n\n\n\\bigskip\nThe following Lemma will play an important  role in the proof of Theorem 4:\n\\smallskip\n\\proclaim Lemma 5. Let $f:M\\to {\\hbox{\\mathbold\\char82}}$ be measurable, and $E\\subseteq M$ measurable with $0<\\mu(E)<\\mu(M)$. Let \n$$\\wtilde f(y)=\\cases{\\displaystyle\\mathop{\\rm {ess\\hskip.2em sup}}\\limits_{z\\in M\\setminus E} |f(z)|& if $\\,y\\in E$\\cr\n f(y) & if $\\,y\\in M\\setminus E$.\\cr}$$\nThen\n$$\\wtilde f^*(t)\\ge f^*(t),\\qquad \\mu(E)\\le t\\le \\mu(M).$$\nMoreover,\n$$\\int_{M\\setminus E} |\\wtilde f|^\\beta=\\int_{\\mu(E)}^{\\mu(M)} [\\wtilde f^*(t)]^\\beta dt.$$\n\\par\\medskip\n\\pf Proof of Lemma 5. Suppose first that $f$ is essentially bounded on $M\\setminus E$ (actually this is all we need for the proof of Theorem 4). If $s_0=\\displaystyle\\mathop{\\rm {ess\\hskip.2em sup}}\\limits_{z\\in M\\setminus E} |f(z)|$, then \n$|f|\\le s_0$ a.e. on $M\\setminus E$, so that  $m(f,s_0)\\le \\mu(E)$. This implies\nthat for $ \\mu(E)\\le t\\le  \\mu(M)$ $$f^*(t)\\le f^*\\big(\\mu(E)\\big)=\\inf\\{s:m(f,s)\\le \\mu(E)\\}\\le s_0$$\nwhich proves the claim if $\\wtilde f^*(t)=s_0$ (it cannot be $>s_0)$. On the other hand\n$$\\{y:\\,|\\wtilde f(y)|>s\\}=\\{y:\\,|f(y)|>s\\}\\cup E,\\,\\qquad 0<s<s_0$$\n hence $m(\\wtilde f,s)\\ge m(f,s)$ if $0<s<s_0$, so that if $\\wtilde f^*(t)=\\inf\\{s: m(\\wtilde f,s)\\le t\\}<s_0$, then \n$$\\wtilde f^*(t)=\\inf\\{s<s_0: m(\\wtilde f,s)\\le t\\}<s_0\\}\\ge\\inf\\{s<s_0: m(f,s)\\le t\\}\\ge f^*(t).$$\n\nFor the last statement, note that $\\wtilde f^*(t)=s_0$ for $0<t<\\mu(E)$ (indeed $m(\\wtilde f,s)=0$ for $s\\ge s_0$, and for any $s<s_0$ we have $|\\wtilde f(y)|>s$ on $E$  and on a set of positive measure inside $M\\setminus E$, i.e. $m(\\wtilde f,s)>\\mu(E)$, for $s<s_0$). Hence\n$$\\int_M  |\\wtilde f|^\\beta=\\int_{\\mu(E)}^{\\mu(M)} [\\wtilde f^*(t)]^\\beta dt+s_0^\\beta\\mu(E)$$\nbut also \n$$\\int_M |\\wtilde f|^\\beta =\\int_{M\\setminus E}|\\wtilde f(y)|^\\beta d\\mu(y)+s_0^\\beta \\mu(E).$$\nA standard approximation argument (truncation and the monotone convergence theorem) completes the proof for general $f$.\n$$\\eqno\/\\!\/\\!\/$$\n\\pf Proof of Theorem 4.  Define for $m\\in {\\hbox{\\mathbold\\char78}}}\\def\\S{{\\cal S} $ \n$$\\Phi_m(y)=K(x_m,y)|K(x_m,y)|^{\\beta\/\\beta'-1}\\chi_{M\\setminus E_m}^{}(y)$$\n$$F_m=\\{y: \\,|K(x_m,y)|>m\\}\\subseteq E_m$$\nso that, by (41) and a), and since $F_m$ is a level set for $|K(x_m,y)|$, \n$$\\eqalign{\\|\\Phi_m\\|_{\\beta'}^{\\beta'}&=\\int_{M\\setminus E_m}|K(x_m,y)|^{\\beta}d\\mu(y)\\le \\int_{M\\setminus F_m}|K(x_m,y)|^{\\beta}d\\mu(y)=\n\\int_{\\mu(F_m)}^{\\mu(M)} [k^*(x_m,\\cdot)(t)]^\\beta dt\\cr&\\le \\int_{\\mu(F_m)}^{\\mu(M)} A \\Big(1+C(1+|\\log t|)^{-\\gamma}\\Big)\\,{dt\\over t}=A\\,\\log{1\\over\\mu(F_m)}+C\\le A\\,\\log{1\\over\\mu(E_m)}+C'\\cr}$$\n(the last inequality follows from the assumptions a) and  c)).\nOn the other hand, if we define \n $$\\wtilde k_m(y)=\\cases{{\\displaystyle{\\rm ess} \\!\\!{\\sup_{\\!\\!\\!\\!\\!\\!z\\in M\\setminus E_m} }}|K(x_m,z)| & if $\\,y\\in E_m$\\cr\\cr |K(x_m,y)| &if $\\,y\\in M\\setminus E_m$\\cr}$$\nthen by Lemma 5 we have $\\wtilde k_m^*(t)\\ge k^*(x_m,\\cdot)(t)$, for $\\mu(E_m)\\le t\\le \\mu(M)$, so that (by (42))\n$$\\eqalignno{\\|\\Phi_m\\|_{\\beta'}^{\\beta'}&=\\int_{M\\setminus E_m}\\!\\!|K(x_m,y)|^{\\beta}d\\mu(y)= \\int_{M\\setminus E_m}\\!\\!|\\wtilde k_m(y)|^{\\beta}d\\mu(y)\n=\\int_{\\mu(E_m)}^{\\mu(M)} [\\wtilde k_m^*(x_m,\\cdot)(t)]^\\beta dt\\cr&\\ge \\int_{\\mu(E_m)}^{\\mu(M)} A \\Big(1-C(1+|\\log t|)^{-\\gamma}\\Big)\\,{dt\\over t}= A\\,\\log{1\\over\\mu(E_m)}-C.&(50)\\cr} $$\nwhich gives (45).\nNow, for $x\\in B_m$, using (50) and (43)\n$$\\eqalign{&T\\Phi_m(x)=\\int_M K(x,y)\\cdot\\Phi_m(y)\\,d\\mu(y)=\\int_{M\\setminus E_m} K(x,y)\\cdot K(x_m,y)\\,|K(x_m,y)|^{\\beta\/\\beta'-1}\nd\\mu(y)\\cr&=\\int_{M\\setminus E_m} |K(x_m,y)|^{1+\\beta\/\\beta'}d\\mu(y)+\\int_{M\\setminus E_m}\\Big(K(x,y)- K(x_m,y)\\Big)\\cdot K(x_m,y)|K(x_m,y)|^{\\beta\/\\beta'-1}\\,d\\mu(y)\\cr& \\ge A\\log{1\\over \\mu(E_m)}- C\n\\cr}$$\nwith $C$ independent of $m$. Hence, if $\\,\\wtilde \\Phi_m=\\Phi_m  \\|\\Phi_m\\|_{\\beta'}^{-1}$\nand  $x\\in B_m$\n$$ T\\wtilde \\Phi_m(x)\\ge {A\\log\\displaystyle{1\\over \\mu(E_m)}- C\\over \\Big(A\\,\\log\\displaystyle{1\\over\\mu(E_m)}\\Big)^{1\/\\beta'}+O(1)}=\\Big(A\\log{1\\over \\mu(E_m)}\\Big)^{1\/\\beta}+O(1).$$\nFinally, if $\\alpha> \\displaystyle{\\beta_0\\over A\\beta}$\n$$\\eqalign{\\int_N \\exp\\Big[\\alpha |T\\wtilde \\Phi_m(x)|^\\beta\\Big]d\\nu(x)&\\ge \\int_{B_m}e^c \\exp\\Big[\\alpha A\\log{1\\over \\mu(E_m)}\\Big] \\,d\\nu(x)\\cr& =e^c\\nu(B_m)\\big(\\mu(E_m)\\big)^{-\\alpha A}\\ge C m^{-\\beta_0+\\alpha A \\beta}\\to+\\infty.\\cr}$$\n$$\\eqno\/\\!\/\\!\/$$\n\\bigskip\n\\noindent{\\bf Remark 1.}  It is clear from the proof just shown that Theorem 4 holds almost verbatim when $K$ is complex-valued and $T$  acts on complex-valued functions. The functions $\\Phi_m$  need only to be replaced by\n$$\\Phi_m(y)=\\overline {K(x_m,y)}\\,|K(x_m,y)|^{\\beta -2}\\chi_{M\\setminus E_m}^{}(y)$$\n\\vskip1em\n\\noindent{\\bf 3. Sharpness in $\\gamma$}\n\\bigskip\nIn this section we show that if $\\gamma\\le 1$ in (13) then  the conclusion of Theorem 1 is in general false. We do this by considering the simplest setting,  namely $N=M=B(0,1)=\\{x\\in\\R^n}\\def\\B{{\\cal B}:\\,|x|\\le1\\}$, $0<d<n,\\,\\beta'={\\displaystyle {n\\over d}},\\, \\beta=\\beta_0={\\displaystyle{n\\over n- d}}$\n$$T_ d f(x)=\\int_{B(0,1)} |x-y|^{ d-n}\\bigg(1+{1\\over 1+\\big|\\log|x-y|\\big|}\\bigg) f(y)dy\\,\\qquad f\\in L^{\\beta'}\\big(B(0,1)\\big).$$\nAs we mentioned previously, in this case $A=\\displaystyle{\\omega_{n-1}\\over n}.$ For $0<r<1$ consider $f_r(x)=|x|^{- d}\\chi_{\\{r\\le|x|\\le1\\}}^{}(x)$, so that $\\|f_r\\|_{\\beta'}^{}=\\big(\\omega_{n-1}\\log\\displaystyle{\\ts {1\\over r}}\\big)^{1\/\\beta'}$. Letting $\\wtilde f_r(x)=f_r(x)\\|f_r\\|_{\\beta'}^{-1}$ we obtain\n$$\\eqalign{\\big(\\omega_{n-1}&\\log\\displaystyle{\\ts {1\\over r}}\\big)^{1\/\\beta'}T_ d\\wtilde f_r(x)=\\int_r^1s^{n-1- d} ds\\int_{S^{n-1}} \n\\big||x|e_1-s\\sigma\\big|^{ d-n}\\bigg(1+{1\\over1+\\Big|\\log\\big||x|e_1-s\\sigma\\big|\\Big|}\\bigg)d\\sigma \\cr\n&=\\int_{|x|}^{|x|\/r} {dt\\over t}\\int_{S^{n-1}} \n|te_1-\\sigma|^{ d-n}\\bigg(1+{1\\over1+\\big|\\log|te_1-\\sigma|+\\log|x|-\\log t\\big|}\\bigg)d\\sigma\n\\cr\n& =\\int_{|x|}^{|x|\/r} {dt\\over t}\\int_{S^{n-1}} \n\\bigg(1+{1\\over1+\\big|\\log|te_1-\\sigma|+\\log|x|-\\log t\\big|}\\bigg)d\\sigma+\\cr&\\hskip2em+ \\int_{|x|}^{|x|\/r} {dt\\over t}\\int_{S^{n-1}} \n\\big(|te_1-\\sigma|^{ d-n}-1\\big)\\bigg(1+{1\\over1+\\big|\\log|te_1-\\sigma|+\\log|x|-\\log t\\big|}\\bigg)d\\sigma.\\cr\n}$$\n\n\\noindent If $|x|\\le r\/2$ and  $|x|\\le t\\le |x|\/r$, then $\\log{1\\over2}\\le \\log|te_1-\\sigma|\\le\\log{3\\over2}$ and $\\big||te_1-\\sigma|^{ d-n}-1\\big|\\le Ct,$ \nso that the second  term above is bounded, for $|x|\\le r\/2$, and \n$$\\eqalign{T_ d \\wtilde f_r(x)&\\ge{\\omega_{n-1}^{1\/\\beta}\\over \\big(\\log{\\ts {\\ts {1\\over r}}}\\big)^{1\/\\beta'}}\\bigg[\\int_{|x|}^{|x|\/r}\\bigg(1+{1\\over 1+\\log2+\\log t-\\log|x|}\\bigg){dt\\over t}-C\\bigg]\\cr&\\hskip-3em\\ge {\\omega_{n-1}^{1\/\\beta}\\over \\big(\\log{\\ts {1\\over r}}\\big)^{1\/\\beta'}}\\bigg[\\log{\\ts {1\\over r}}+\\log\\bigg(1+{\\log{\\ts {1\\over r}}\\over1+\\log 2}\\bigg)-C\\bigg]\\ge \\big(\\omega_{n-1}\\log{\\ts {1\\over r}}\\big)^{1\/\\beta}\\bigg[1+{\\log \\log{\\ts {1\\over r}}\\over2\\log{\\ts {1\\over r}}}\\bigg]\n\\cr}$$\nfor $0<r\\le r_0<1$, with $r_0$ small enough. Finally, for $r\\le r_0$ \n$$\\eqalign{\\int_{B(0,1)} \\exp\\bigg[{n\\over \\omega_{n-1}}&\\big(T_ d \\wtilde f_r\\big)^\\beta\\bigg] dx\\ge\\int_{B(0,r\/2)}\n\\exp\\bigg[n\\log{1\\over r}\\bigg(1+{\\log \\log{\\ts {1\\over r}}\\over2\\log{\\ts {1\\over r}}}\\bigg)^\\beta\\bigg]dx\\cr\n&\\ge {\\omega_{n-1}\\over n}\\Big({r\\over2}\\Big)^n r^{-n} \\exp\\Big[{\\ts{1\\over2}}n\\beta\\log\\log{1\\over r}\\Big]={\\omega_{n-1}\\over n}\\, 2^{-n} \\big(\\log\\ts{1\\over r}\\big)^{n\\beta\/2}\\to+\\infty,\\cr\n}$$\nas $r\\to0$.\n\n\n\n\\bigskip\n\\bigskip\n\\centerline{\\bf PART II: APPLICATIONS}\n\\bigskip\\bigskip\n\\noindent{\\bf 4. Adams and Moser-Trudinger trace inequalities}\n\\bigskip\n\nAs a first illustration of our theorems we give a simultaneous extension of Adams' original results in [Ad], and Cianchi's recent sharp Moser-Trudinger trace inequality in [Ci1]\n$$\\int_\\Omega \\exp\\bigg[\\lambda\\omega_{n-1}^{1\\over n-1}\\bigg({|u(x)|\\over \\|\\nabla u \\|_{L^n(\\Omega)}}\\bigg)^{n\\over n-1}\\bigg]\\,d\\nu(x)\\le C\\eqno(51)$$\nvalid for all $u\\in W_0^n(\\Omega)$, where $L^n(\\Omega)$ and $W_0^n(\\Omega)$ are w.r. to the Lebesgue measure, and where $\\nu$ denotes a positive Borel measure on $\\Omega$ such that \n$$\\exists \\lambda\\in(0,n]\\,\\, {\\rm and }\\,\\, r_0>0\\,\\,:\\,\\,\\nu\\Big(B(x,r)\\cap \\Omega\\Big)\\le C r^\\lambda,\\qquad \\forall x\\in {\\hbox{\\mathbold\\char82}}^n,\\,\\, \\forall r\\in(0,r_0].\\eqno(52)$$\nHere and throughout the rest of this work \n$$B(x,r)=\\{ y\\in{\\hbox{\\mathbold\\char82}}^n: |y-x|<r\\}.$$\nNote that (51) is itself an extension of Adams' original result when $\\lambda=n$, in the special case of the gradient. Cianchi's results are given in the slightly more general framework of Lorentz-Sobolev spaces.\n\n\nAs an immediate consequence of our general theorem we will now derive  Cianchi's result, for arbitrary powers of the Laplacian and their gradients.   In other words, we will   extend Adams' result (2), in the case of measures satisfying (52). \n\n\nIn the following,  by  $L^p(\\Omega)$ we mean the space of Lebesgue measurable functions $f$  on $\\Omega$ such that $\\|f\\|_p:=\\Big(\\int_\\Omega |f|^pdx\\Big)^{1\/p}<\\infty$.\nAlso,  $\\Delta$ will always denote the positive\nLaplacian on $\\R^n}\\def\\B{{\\cal B}$. The fractional powers of $\\Delta$ are defined on the Schwarz \nclass $\\S$ as\n$$(\\Delta^{ d\/2} \\phi)\\;{\\widehat{\\vphantom{S}}}\\;(\\xi)=(\\;2\\pi|\\xi|\\;)^ d\\;\\widehat\\phi(\\xi)\\,,\n\\qquad\\phi\\in \\S.$$\nHere $ d\\in{\\hbox{\\mathbold\\char82}}$ and the Fourier transform is defined as  $\\;\\widehat\\phi(\\xi)=\\int_{\\sR^n}}\\def\\bSn{{\\bS}^n \ne^{-2\\pi i x\\cdot\\xi} \\phi(x)dx$. By duality we can extend $\\Delta^{ d\/2} $ to an operator\nacting on $\\S'$, the space of tempered distributions.\n\nWhen $0< d<n$ and $f\\in L^p(\\R^n}\\def\\B{{\\cal B})$, $p\\ge1$, the equation $\\Delta^{ d\/2} \\phi=f$ has a unique \nsolution in $L^q$, with $q^{-1}=p^{-1}- d\/n$. This solution is given\nexplicitly in terms of  the Riesz potential \n$$\\Delta^{- d\/2}f(x)=c_ d\\int_{\\sR^n}}\\def\\bSn{{\\bS}^n |x-y|^{ d-n}f(y)dy\\eqno(53)$$\nwith\n$$c_ d={\\Gamma\\big({n- d\\over2}\\big)\\over 2^ d\\pi^{n\/2}\\Gamma\\big({ d\\over2}\\big)}.\\eqno(54)$$ \n\nIn other words, the distributional Fourier transform of the RHS\nof (53) coincides with $(2\\pi|\\xi|)^{- d}\\widehat f(\\xi)$. \n\nThe usual Sobolev space on ${\\hbox{\\mathbold\\char82}}^n$ is denoted by $W^{ d,p}(\\R^n}\\def\\B{{\\cal B})$ and the space $W_0^{ d,p}(\\Omega)$ is the closure of $C_0^\\infty(\\Omega)$ in $W^{ d,p}(\\R^n}\\def\\B{{\\cal B})$.\n\\medskip\n\n\\proclaim Theorem 6. Let $\\Omega$ be open and bounded on $\\R^n}\\def\\B{{\\cal B}$, $n\\ge3$, and let $\\nu$ be a positive Borel  measure  on $\\Omega$ satisfying (52). For $0< d<n$, $ d\\in{\\hbox{\\mathbold\\char78}}}\\def\\S{{\\cal S} $, let $p=\\displaystyle{n\\over d}$ and  $\\,\\displaystyle{{1\\over p}+{1\\over p'}=1}$. Then, there exists $C$ such that\n$$\\int_\\Omega \\exp\\bigg[{\\lambda\\,c_ d^{-p'}\\over \\omega_{n-1}}\\bigg({|u(x)|\\over\\|\\Delta^{d\/2}u\\|_p}\\bigg)^{p'}\\bigg]d\\nu(x)\\le C,\\qquad  d {\\hbox{ even}}\\eqno(55)$$\nand\n$$\\int_\\Omega \\exp\\bigg[{\\lambda\\over\\omega_{n-1}}\\big(c_{ d+1}(n- d-1)\\big)^{-p'}\\bigg({|u(x)|\\over\\|\\nabla\\Delta^{{ d-1}\\over2}u\\|_p}\\bigg)^{p'}\\bigg]d\\nu(x)\\le C,\\qquad  d {\\hbox{ odd}}\\eqno(56)$$\nfor all $u\\in W_0^{ d,p}(\\Omega)$. The constants appearing inside the exponents in (55) and (56) are sharp, provided there exists $x_0\\in\\Omega$ such that $\\nu\\big(B(x_0,r)\\cap \\Omega\\big)\\ge C_1 r^\\lambda$ for $0< r\\le r_1$, some $r_1,C_1>0$.\n  \\par\n\n\\bigskip\n\nWhen $\\lambda=n$ and $ d=m$ one recovers the constants $\\beta(m,n)$ appearing in [Ad1]. When $ d=1$  the constant in \n(56) coincides with that of Cianchi, for $0<\\lambda\\le n$. \n\\medskip\nIt is clear that it is enough to  prove the theorem if $u$ is smooth  with compact support inside $\\Omega$. Secondly, for $ d$ even\n$$u(x)=c_ d\\int_{\\Omega} |x-y|^{ d-n} \\Delta^{ d\/2}u(y)dy$$ \nand for $ d$ odd\n$$u(x)=c_{ d+1}(n- d-1)\\int_\\Omega |x-y|^{ d-n-1}(x-y)\\cdot\\nabla \\Delta^{{ d-1\\over2}}u(y)dy,\\eqno(57)$$ \nand therefore the inequalities of Theorem 6 are instant consequences of the following:\n\n\\proclaim Theorem 7. Let $\\Omega$ be open and bounded on $\\R^n}\\def\\B{{\\cal B}$, $n\\ge1$, and let $\\nu$ be a positive Borel  measure  on $\\Omega$ satisfying (52). For $0< d<n$, let $p=\\displaystyle{n\\over d}$ and  $\\,\\displaystyle{{1\\over p}+{1\\over p'}=1}$. Define for $f:\\Omega\\to{\\hbox{\\mathbold\\char82}}$, $f\\in L^p(\\Omega)$\n$$Tf(x)=\\int_\\Omega |x-y|^{ d-n} f(y)dy$$\nand for $F:\\Omega\\to{\\hbox{\\mathbold\\char82}}^n$, $F\\in L^p(\\Omega)$\n$$T_1F(x)=\\int_\\Omega |x-y|^{ d-n-1}(x-y)\\cdot F(y)dy.$$\nThen, there exists $C$ such that\n$$\\int_\\Omega \\exp\\bigg[{\\lambda\\over \\omega_{n-1}}\\bigg({|Tf(x)|\\over\\|f\\|_p}\\bigg)^{p'}\\bigg]d\\nu(x)\\le C\\eqno(58)$$\nfor every $f\\in L^p(\\Omega)$, and\n$$\\int_\\Omega \\exp\\bigg[{\\lambda\\over\\omega_{n-1}}\\bigg({|T_1F(x)|\\over\\|F\\|_p}\\bigg)^{p'}\\bigg]d\\nu(x)\\le C\\eqno(59)$$\nfor every $F\\in L^p(\\Omega)$. The constant $\\displaystyle{\\lambda\\over \\omega_{n-1}}$ in (58) and (59) is sharp, provided there exists $x_0\\in\\Omega$ such that $\\nu\\big(B(x_0,r)\\cap \\Omega\\big)\\ge C_1 r^\\lambda$, for $0< r\\le r_1$, some $r_1,C_1>0$.\n \\par\n\n\\pf  Proof.\nIt is easy to check that if $k(x,y)=|x-y|^{ d-n}$ then for large $s$\n\n$$m(k_1^*,s)={\\omega_{n-1}\\over n}s^{-{n\\over n- d}}$$\nand, using (52),\n$$m(k_2^*,s)\\le C s^{-{\\lambda\\over n- d}},$$\nso that  Theorems 1-1' immediately imply (58), (59). To verify sharpness, according to Theorem 4, (and Remark 1 following it) first assume WLOG that $x_0=0\\in \\Omega$,  then take  $x_m=0\\in\\Omega,$ and   $m,R$ large enough so that \n\n$$ \\{y\\in\\Omega:\\,|K(0,y)|>m\\,\\}= B(0, m^{-p'\/n})\\subseteq\\Omega\\subseteq B(0,R),\\eqno(60)$$\nand let $$r_m=m^{-p'\/n},\\;\\; E_m=B(0,r_m)\\;\\;B_m=B(0,\\ts{1\\over2}r_m)\\eqno(61)$$\nwith either $K(x,y)= |x-y|^{ d-n} $ or $K(x,y)= |x-y|^{ d-n-1}(x-y)$. Conditions a), b), c) of Theorem 4 are met, with $\\beta=n\/(n- d)$ and $\\beta_0=\\lambda\/(n- d)$, so all we need to check is d), i.e.\n\n$$\\int_{\\Omega\\setminus E_m} \\big|\\big(K(x,y)- K(0,y)\\big)\\cdot K(0,y)\\big|\\, |K(0,y)|^{p'-2} dy\\le C\\,,\\qquad  |x|\\le {r_m\\over2}$$\nfor either kernel. If $K(x,y)= |x-y|^{ d-n} $ we need to check\n\n$$\\sup_{|x|\\le r_m\/2}\\;\\int_{r_m\\le |y|\\le R} |y|^{- d}\\Big||x-y|^{ d-n}-|y|^{ d-n}\\Big|dy\\le C\\eqno(62)$$\nfor some $C$ independent of $m$, but this  estimate is an   immediate consequence of \n$$|x-y|^{ d-n}\\le |y|^{d-n}\\Big|{x\\over|y|}-{y\\over|y|}\\Big|^{ d-n}\\le 2^{n- d} |y|^{ d-n}\\eqno(63)$$\nand\n$$ |y|^{- d}\\Big||x-y|^{ d-n}-|y|^{ d-n}\\Big|\\le C|y|^{-n}\\bigg|1-\\Big|{x\\over|y|}-{y\\over|y|}\\Big|^{n- d}\\bigg|\\le C|x||y|^{-n-1},\\eqno(64)$$\nboth valid for any $ d<n$ (even negative) and $|x|\\le{r_m\\over2},\\;|y|\\ge r_m$. \n\\medskip\nIf instead $K(x,y)= |x-y|^{ d-n-1}(x-y)$ then we are reduced to\n$$\\sup_{|x|\\le r_m\/2}\\;\\int_{r_m\\le |y|\\le R} \\Big||x-y|^{ d-n-1}(x\\cdot y-|y|^2)+|y|^{d-n+1}\\Big|\\,|y|^{- d-1}dy\\le C$$\nwhich is implied by\n$$\\sup_{|x|\\le r_m\/2}\\;\\int_{r_m\\le |y|\\le R} |x|\\,|y|^{- d}\\,|x-y|^{d-n-1}dy\\le C$$\n$$\\sup_{|x|\\le r_m\/2}\\;\\int_{r_m\\le |y|\\le R} \\Big||x-y|^{ d-1-n}-|y|^{d-1-n}\\Big|\\,|y|^{-( d-1)}dy\\le C,$$\nand both of these are also easy consequences of (63) and (64). \n\n$$\\eqno\/\\!\/\\!\/$$\n\n\n\\noindent{\\bf Proof of Theorem 6}. The inequalities of Theorem 6 follows from the formulas\n$$u(x)=Tf(x)=\\int_{\\Omega} |x-y|^{d-n} f(y) dy,\\qquad f=c_d\\Delta^{d\/2}u$$\nfor $d$ even, and \n$$u(x)=T_1F(x)=\\int_\\Omega |x-y|^{d-n-1}(x-y)\\cdot F(y)dy,\\qquad F(y)=c_{d+1}(n-d-1)\\nabla\\Delta^{d-1\\over2}u$$\n for $d$ odd.\n\nAs far as the sharpness statement, the proof we give below is a slight modification of  Adams' original method. We include some details here since similar formulas  will be used later in the proof  of  Theorem 12. It is possible however to give a proof based on our Theorem 4 by using representation formulas via the Green function of $\\Delta^{d\/2}$ on the unit ball, with zero boundary conditions. Later in theorem 10 we will give a  sharp inequality for more general (scalar) operators, and the proof of sharpness given there is a more direct application of Theorem 4, and applies also to the case $d$ even of the present theorem (see remark 2 after the proof of theorem 10). \n\n\nLet $r_m\\to0^+$ and $B_m=B(0,r_m)\\subseteq \\Omega$, for $m$ large enough.\n\n\n Next, pick any $\\delta\\ge0  $ and  $\\varphi\\in C^\\infty([\\delta,1+\\delta])$ such that  $\\varphi^{(k)}(\\delta)=0$ for $0\\le k\\le \\ell$, some $\\ell\\ge d$, and $\\varphi(1+\\delta)=1+\\delta, \\varphi'(1+\\delta)=1$, $\\varphi^{(k)}(1+\\delta)=0$ for $2\\le k\\le \\ell$.\n\nDefine for $m$ large enough \n$$u_m(y)=\\cases{0 &  for $\\;|y|\\ge e^{-\\delta}$\\cr \\varphi\\Big(\\log\\displaystyle{1\\over |y|}\\Big) & for $\\;e^{-1-\\delta}\\le |y|<e^{-\\delta}$\\cr  \\log\\displaystyle{1\\over|y|} & for $\\;e^{1+\\delta} r_m\\le |y|< e^{-1-\\delta}$\\cr   \\log\\displaystyle{1\\over r_m}-\\varphi\\Big({\\log\\displaystyle{|y|\\over r_m}\\Big)} & for $\\;e^\\delta r_m\\le|y|<e^{1+\\delta} r_m$\\cr \n\\log\\displaystyle{1\\over r_m}& for $\\;|y|<e^\\delta r_m.$\\cr}\\eqno(65)$$ \n\n Then, if we pick $\\delta$ large enough we have $u_m\\in W_0^{d,p}(\\Omega)$ and it is easy to check that for $d$ even\n$$\\Delta^{d\/2}u_m(y)=\\cases{ \\displaystyle{|y|^{-d}\\over \\omega_{n-1}c_d}& if \n$e^{1+\\delta} r_m<|y|< e^{-1-\\delta}$\\cr\\cr O(1)|y|^{-d} & if $e^\\delta r_m<|y|<e^{1+\\delta} r_m$\\cr\\cr O(1) & otherwise\\cr}$$\nand\n$$\\|\\Delta^{d\/2}u_m\\|_p^{p'}={1\\over\\omega_{n-1} c_d^{p'}}\\Big(\\log{1\\over r_m}\\Big)^{p'\/p}+O(1).$$\nThis means that if  $$\\int_\\Omega \\exp\\bigg[\\alpha \\,\\bigg({|u(x)|\\over\\|\\Delta^{d\/2}u\\|_p}\\bigg)^{p'}\\bigg]d\\nu(x)\\le C$$\nholds for all $u\\in W^{d,p}(\\Omega)$, then according to (49)\n$$\\alpha\\le \\liminf_n\\Bigg({\\|\\Delta^{d\/2} u_m\\|_p\\over\\log(1\/r_m)}\\bigg)^{p'}\\log{1\\over\\nu(B_m)}={\\lambda\\over \\omega_{n-1} c_d^{p'}}.$$\nThe proof in the case $d$ odd is completely similar, and based on the identity \n$$\\nabla\\Delta^{d-1\\over2}\\log{1\\over|y|}={(d-1)y|y|^{-d-1}\\over \\omega_{n-1}c_{d-1}}=-{y|y|^{-d-1}\\over \\omega_{n-1}(n-d-1)c_{d+1}}.$$\n$$\\eqno\/\\!\/\\!\/$$\n\nWe remark here that the above theorem can be formulated and proved  in the setting of compact Riemannian manifolds, in the same spirit as in [F]. The point is that such theorem holds for any pseudodifferential operator of order $d$, whose leading symbol is $\\Delta^{ d\/2}$ (see next section for even a more a general result). For such operators the fundamental solution $k(P,Q)$ satisfies locally\n$$k(P,Q)=c_ d d(P,Q)^{ d-n}\\Big(1+O\\big(d(P,Q)^{\\epsilon}\\def\\part{\\partial_t}\\big)\\Big)$$\nfor some $\\epsilon}\\def\\part{\\partial_t>0$, where $P,Q$ are points on the manifold, and $d(P,Q)$ is their Riemannian distance. Under these conditions it is easy to check that\n$$k_1^*(t)={\\omega_{n-1}\\over n}(c_ d)^{-p'}t^{-1\/p'}\\Big(1+O\\big(t^{-1\/p'+\\epsilon}\\def\\part{\\partial_t}\\big)\\Big)$$\nfor small $t$, and it is clear that the estimate $k_2^*(t)\\le C t^{-n\/(\\lambda p')}$ would follow if the underlying  Borel measure $\\nu$ satisfies $\\nu\\big(B(P,r)\\big)\\le Cr^\\lambda$, for small geodesic balls $B(P,r)$. These facts, and similar ones for vector-valued operators, imply  inequalities  such as  those of Theorems 6 and 7, and the sharpness statements are proven in essentially the same manner.\n\n\nIt would also be possible to extend this theorem to general Lorentz-Sobolev space, in the same spirit as in [Ci1], with  suitable and  slightly more general versions of our theorems 1,1' and 4, which for simplicity we only treated in the  $L^p$ setting.\n\n\nFinally, we wish to remark that our proof of Theorem 6 is of a somewhat different nature than the  one given by Cianchi in [Ci1]. In the special case $ d=1$ Cianchi started by applying the Sobolev inequality\n$$\\|\\Psi\\|_{L^{\\lambda p\\over n-p}(\\Omega,d\\nu)}^{}\\le C\\|\\nabla \\Psi\\|_{L^p(\\Omega)}^{}\\eqno(66)$$\nfor some suitable $p<n$, with $\\Psi(x)=\\exp\\Big[\\displaystyle{n-p\\over p}\\,\\omega_{n-1}^{1\\over n-1}\\, |u|^{n'}(x)\\Big]-1$. Note that (66) holds on $W_0^{1,p}$ and it is a special case of a general Sobolev inequality derived by Adams [Ad2], [Ad3]. Next, the fact that $\\nabla e^u=e^u \\nabla u$ combined with clever use of rearrangement inequalities and O'Neil's\/Adams-Garsia's lemmas allowed Cianchi to obtain the result. However, it is unclear to us how to efficiently apply this stategy to arbitrary order derivatives.\n\n \n\\vskip1em\\eject\n\\noindent{\\bf 5.  Sharp Moser-Trudinger inequalities for general elliptic differential operators}\n\\bigskip\n\nIn this section we extend Adams' original inequality to general elliptic differential operators on $\\R^n}\\def\\B{{\\cal B}$. We then specialize to particular cases. Some of these results can perhaps be further extended to suitable elliptic pseudodifferential operators, operators on manifolds, or even non-elliptic operators, but we will not treat such cases here.  \n\nThe structure of the fundamental solution of general elliptic operators is best explained by use of the pseudodifferential calculus, which we now very briefly recall.\n\\smallskip\\noindent{\\bf Note.} \nThroughout this section all  functions will be {\\it complex-valued}, unless otherwise specified.\n\\smallskip\n\\def{$\\Psi$DO} {{$\\Psi$DO} }\nA pseudodifferential ({$\\Psi$DO} ) operator of order $d\\in{\\hbox{\\mathbold\\char82}}$\non an open set $U\\subset\\R^n}\\def\\B{{\\cal B}$ \nis an operator $P:C_c^\\infty(U)\\rightarrow C^\\infty(\\R^n}\\def\\B{{\\cal B})$ of type \n$$Pf(x)=\\op(p)f(x):=\\int_{\\sR^n}}\\def\\bSn{{\\bS}^n\\!\\int_{\\sR^n}}\\def\\bSn{{\\bS}^n e^{2\\pi i(x-y)\\cdot\\xi}p(x,\\xi)f(y)dyd\\xi$$\nwhere $p\\in C^\\infty(U\\times\\R^n}\\def\\B{{\\cal B})$, the symbol of $P$, satisfies\n $$|\\partial_x^\\beta \\partial_\\xi^\\alpha p(x,\\xi)|\\le C_{ d,\\beta,K} (1+|\\xi|)^{d-|\\alpha|},\\qquad x\\in K,\\, \\xi\\in\\R^n}\\def\\B{{\\cal B}$$\n for any $K\\subset U$ compact, and any multiindices $\\alpha,\\beta$ (where $C$ is independent of $x,\\,\\xi$), and where $\\partial^{\\alpha}_x=\\partial_{x_1}^{i_1},...\\partial_{x_n}^{i_n}$, if $\\alpha=(i_1,...,i_n)$.\n\nA classical, or polyhomogeneous, $\\Psi$DO of order $d$ is given by a symbol $p(x,\\xi)$ such that\n\n\\smallskip\n\\noindent i) there is a sequence of functions $p_{d-j}(x,\\xi)\\in C^\\infty(U,\\R^n}\\def\\B{{\\cal B})$ which are homogeneous of order $d-j$ in $\\xi$, for $|\\xi|\\ge1$:\n $$p_{d-j}(x,t\\xi)=t^{d-j}p_{d-\nj}(x,\\xi), \\qquad t\\ge1,\\;\\;|\\xi|\\ge1,\\;\\;j=0,1,2... $$\n\n\n\n\\smallskip\\noindent ii) $p$ has the asymptotic expansion\n$p\\sim \\sum_{j=0}^\\infty p_{d-j},$ that is\n$$\\Big|\\partial_x^\\beta \\partial_\\xi^\\alpha \\Big[p(x,\\xi)-\\sum_{j=0}^{N-1} p_{d-\nj}(x,\\xi)\\Big]\\Big|\\le C_{\\alpha,\\beta,K,N} (1+|\\xi|)^{d-|\\alpha|-N},\\qquad x\\in K,\\,\\xi\\in\\R^n}\\def\\B{{\\cal B}$$\nfor any $M\\in {\\hbox{\\mathbold\\char78}}}\\def\\S{{\\cal S} $, any multiindices $\\alpha,\\beta$, and any compact set $K\\subset U$.\n\nThe principal symbol of such $P$ is the function $p_d(x,\\xi)$ and the strictly homogeneous symbol\nis defined by \n$$p_d^0(x,\\xi)=|\\xi|^d p_d(x,{\\xi\/|\\xi|}),\\qquad \\xi\\neq0,$$\nthat is the unique function on ${\\hbox{\\mathbold\\char82}}^n\\setminus \\{0\\}$ which coincides with $p_d$ for $|\\xi|\\ge1$, and which is homogeneous\nor order $d$ in $\\xi\\in \\R^n}\\def\\B{{\\cal B}\\setminus\\{0\\}$.\n\nFrom now on we will consider only classical  {$\\Psi$DO} $\\!\\!$.\nEvery {$\\Psi$DO}  can be written as an integral operator with kernel\n$$K_P(x,y)=\\int_{\\sR^n}}\\def\\bSn{{\\bS}^n e^{2\\pi i(x-y)\\cdot\\xi}p(x,\\xi)d\\xi$$\ndefined in the sense of oscillatory integrals, and $C^\\infty$ off the diagonal. If $K_P$ is $C^\\infty(U\\times U)$ then $K_P$ is called  smoothing (or negligible) operator.\n\n\nFor a classical {$\\Psi$DO}  $T$ of order $d$, with $-n\\le d< 0$  the  Schwarz kernel of $T$ has an expansion \n$$K(x,y)=\\mathop{\\sum}\\limits_{0\\le j<n-d}  k_{d-n+j}(x,x-y)+c(x)\\log|x-y|+O(1)\\eqno(67)$$\nwith $k_{d-n-j}(x,z)$ homogeneous of order $d-n-j$ in $z\\in\\R^n}\\def\\B{{\\cal B}\\setminus\\{0\\}$, $c(x)$ continuous on $U$, and $O(1)$ continuous and bounded on $K\\times K$ for any $K\\subset U$ compact. This fact is standard, (see e.g. [Ca], Theorem 28) and follows  from taking the inverse   Fourier transform of the expansion of the symbol of $T$.\nTo be more specific,  let  $T$ have symbol $q$ with\n$$q\\sim \\sum_{j=0}^\\infty q_{-d-j}\\quad x\\in U,\\, |\\xi|\\ge1,\\eqno(68)$$\nwhere $q_{-d-j}$ are homogeneous of order $-d-j$ for $|\\xi|\\ge 1$, and let  \n${\\cal F} u=\\hat u$ denote  the Fourier transform  on tempered distributions. Then for  $j<n-d$ the $q_{-d-j}^0$ are integrable around the origin, so that  modulo a smooth function ${\\cal F}^{-1}\\big[q_{-d-j}^0(x,\\cdot)\\big](z)$ coincides with\n$$k_{d-n-j}(x,z):={\\cal F}^{-1} \\big[q_{-d-j}^0(x,\\cdot)\\big](z)\\eqno(69)$$\nwhich  is a homogeneous function of order $d-n-j$ in $z$, smooth in $x$ and in $z\\neq 0$. When $j=n-d$ (which can only happen for integer $d$), then one can show (see [Ca], Theorem 27) that \n$${\\cal F}^{-1} \\big[q_{-n}(x,\\xi)](z)=c(x)\\log{1\\over|z|}+O(1)\\eqno(70)$$\nas $z\\to0$, with $c$ continuous, and in fact $c(x)=\\displaystyle{\\int_{S^{n-1}}} q_{-n}^0(x,\\omega)d\\omega$ (the term $O(1)$ is in fact the sum of a  $C^\\infty(\\R^n}\\def\\B{{\\cal B}\\setminus0)$ homogeneous function of degree $0$ and a polynomial). Finally, the inverse Fourier transform of the error term $q-\\sum_{j\\le n-d}q_{-d-j}$ is easily estimated to be $O(1)$. \n\n\nA {$\\Psi$DO}   operator $P$ is elliptic, if\n$$p_d(x,\\xi)\\neq0,\\qquad x\\in U,\\,|\\xi|\\ge 1$$\ni.e. $p_d^0(x,\\xi)\\neq 0$ for $\\xi\\neq0$.  For elliptic operators one can construct a parametrix $G$, i.e. a {$\\Psi$DO}  of order $-d$ such that $GP-I=PG-I$ is smoothing. The  symbol $q(x,\\xi)$ of $G$ has an expansion like (68), where the $q_{-d-j}$  are determined from the symbol of $P$. In particular\n$$ q_{-d}(x,\\xi)={1\\over p_d(x,\\xi)}, \\quad x\\in U,\\, |\\xi|\\ge1$$\nso that the  parametrix admits a kernel expansion (67), with leading term given by \n$$k_{d-n}(x,x-y)={\\cal F}^{-1} \\big[p_{-d}(x,\\cdot)^{-1}\\big](x-y)=|x-y|^{d-n} g\\big(x,{y-x\\over|y-x|}\\big)\\eqno(71)$$\nwith $$g(x,\\omega)=\\bigg({1\\over p_d^0(x,\\cdot)}\\bigg)^\\wedge (\\omega),\\qquad\\omega\\in S^{n-1}\\eqno(72)$$\nthe restriction of $k_{d-n}(x,\\cdot)$ (which is $C^\\infty({\\hbox{\\mathbold\\char82}}-\\setminus\\{0\\})$) to the sphere.\n\n\\medskip\\smallskip\\bigskip\n\\noindent{\\it  Sharp inequalities  for the potential case}\\medskip\n\\medskip\nThe precise asymptotic expansion of the parametrix operator $G$ suggests the following general theorem:\n\\bigskip\n\\proclaim Theorem 8. Let $\\Omega$ be open and bounded on $\\R^n}\\def\\B{{\\cal B}$,  and let $K:\\Omega\\times \\Omega\\to\\overline {\\hbox{\\mathbold\\char82}}$ be  measurable and such that\n$$K(x,y)=g\\Big(x,{y-x\\over|y-x|}\\Big)\\,|x-y|^{ d-n}+O\\big(|x-y|^{ d-n+\\epsilon}\\def\\part{\\partial_t}\\big)\\eqno(73)$$ \nwhere $g:\\Omega\\times S^{n-1}\\to {\\hbox{\\mathbold\\char82}}$ is measurable and bounded, and $\\epsilon}\\def\\part{\\partial_t>0$.  For $0< d<n$, $\\,p=\\displaystyle{n\\over d}$, $\\,\\displaystyle{{1\\over p}+{1\\over p'}}=1$, let, for $x\\in \\Omega$ \n$$Tf(x)=\\int_\\Omega K(x,y)f(y)dy,\\quad f\\in L^p(\\Omega).$$ Then there exists $C>0$ such that \n$$\\int_{\\Omega} \\exp\\bigg[A^{-1}\\bigg({|Tf(x)|\\over \\|f\\|_p}\\bigg)^{p'}\\bigg]\\,dx\\le C\\eqno(74)$$\nfor all $f\\in L^p(\\Omega)$, with \n$$A={1\\over n}\\,\\sup_{x\\in \\Omega}\\int_{S^{n-1}}|g(x,\\omega)|^{p'}d\\omega.\\eqno(75)$$ \n\\eject\nIf the supremum in (75) is attained at some $x_0\\in\\Omega$, and if $g(\\cdot,\\omega)$ is H\\\"older continuous of order $\\sigma\\in (0,1]$ at $x_0$ uniformly w.r. to $\\omega$, i.e. if \n$$|g(x,\\omega)-g(x_0,\\omega)|\\le C|x-x_0|^\\sigma \\qquad |x-x_0|\\le \\delta,\\;\\omega\\in S^{n-1}$$ and  $g(x_0,\\cdot)$ is  H\\\"older continuous of order $\\sigma$ on $S^{n-1}$\nthen the constant $A^{-1}$ in (74) is sharp. In particular, there is  a suitable sequence  $r_m\\to 0$ such that if   $\\,E_m=B(x_0, r_m)\\subseteq \\Omega$ and  $\\Phi_m(y)=K(x_0,y)|K(x_0,y)|^{p'-2}\\chi_{\\Omega\\setminus E_m}^{}(y)$, then $\\Phi_m\\in\n L^{p}$ and\n$$\\lim_{m\\to\\infty}\\int_\\Omega \\exp \\bigg[\\alpha\\,\\bigg({|T\\Phi_m|\\over\\|\\Phi_m\\|_{p}^{}}\\bigg)^{p'}\\,\\bigg]dx=+\\infty,\n\\qquad \\forall \\alpha>{1\\over A}.$$\n\n\n\n\\par\n\\medskip\n\\noindent {\\bf Remarks.}\\smallskip\\noindent {\\bf 1.} Cohn and Lu were the first to consider Adams inequalities for potentials of simpler type $g(y\/|y|)|y|^{d-n}$, and the analogous version on the Heisenberg group ([CoLu1]).\\smallskip \\noindent {\\bf 2.} The H\\\"older continuity  condition on $g$ can be relaxed to an integral condition similar to that used in [CoLu1].\n\\medskip\n\n In view of Theorems 1 and 4 it is clear that to prove Theorem 8 it would essentially suffice to estimate the distribution function of the kernel $K$. This is done in the following lemma:\n\\proclaim Lemma 9. Suppose that $K$  is as in Theorem 8, satisfying (73) with $g$ bounded and measurable. Then for $s>0$ \n$$\\sup_{x\\in\\Omega} \\,|\\{y\\in\\Omega: \\,|K(x,y)|>s\\}|\\le   A s^{-p'}+O(s^{-p'-\\sigma})\\eqno(76)$$\n for suitable $\\sigma>0$, with $A$ as in (75), with equality if  the sup in (75) is attained in $\\Omega$. Moreover,   \n$$\\sup_{y\\in\\Omega}\\,|\\{x\\in\\Omega: \\,|K(x,y)|>s\\}|\\le C  s^{-p'}.\\eqno(77)$$\n\\par\\smallskip\n\\noindent {\\bf Note.} A similar lemma was proved in [BFM], Lemma 2.3, for kernels in the CR sphere. \n\\smallskip\n\n\n\\bigskip\n\\pf Proof of Lemma 9. From now on we will use the notation \n$$y^*={y\\over|y|}.$$\n The hypothesis implies \n$$|K(x,y)|\\le|g(x,(y-x)^*)|\\,|x-y|^{ d-n}+C|x-y|^{ d-n+\\epsilon}\\def\\part{\\partial_t}$$\nso that for any $x\\in \\Omega$\n$$m_x(s):=|\\{y\\in\\Omega: \\,|K(x,y)|>s\\}|\\le |\\{y\\in\\R^n}\\def\\B{{\\cal B}: \\,|g(x,y^*)|\\,|y|^{ d-n}+C|y|^{ d-n+\\epsilon}\\def\\part{\\partial_t}>s\\}|$$\nand since\n$$|g(x,y^*)|\\,|y|^{ d-n}+C|y|^{ d-n+\\epsilon}\\def\\part{\\partial_t}>s\\;\\;\\Longrightarrow \\;\\; |y|\\le s^{-p'\/n}\\big(|g(x,y^*)|+C|y|^\\epsilon}\\def\\part{\\partial_t\\big)^{p'\/n}\\le C s^{-p'\/n}$$\nthen\n$$m_x(s)\\le{s^{-p'}\\over n}\\int_{S^{n-1}}\\big(|g(x,y^*)|+Cs^{-\\epsilon}\\def\\part{\\partial_t p'\/n}\\big)^{p'}dy^*$$\nwhich implies (76). \nSuppose that for some $x_0\\in \\Omega$\n$$A={1\\over n}\\int_{S^{n-1}} |g(x_0,\\omega)|^{p'}d\\omega$$\nand WLOG we can assume that $x_0=0$.\nSince \n$$|K(0,y)|\\ge|g(0,y^*)|\\,|y|^{ d-n}-D|y|^{ d-n+\\epsilon}\\def\\part{\\partial_t}$$\nfor some $D>0$, then\n$$\\eqalign{m_0(s)&\\ge |\\{y\\in\\Omega: \\,|g(0,y^*)|\\,|y|^{ d-n}-D|y|^{ d-n+\\epsilon}\\def\\part{\\partial_t}>s\\}|\\cr&=|\\{y\\in\\Omega: \\,|y|<s^{-p'\/n}\\big(|g(0,y^*)|-D|y|^{\\epsilon}\\def\\part{\\partial_t}\\big)^{p'\/n}\\}|.\\cr}$$\nBut\n$$|y|<s^{-p'\/n}\\big(|g(0,y^*)|-D|g(0,y^*)|^{\\epsilon}\\def\\part{\\partial_t p'\/n}s^{-\\epsilon}\\def\\part{\\partial_t p'\/n}\\big)^{p'\/n}\\Longrightarrow |y|<s^{-p'\/n}\\big(|g(0,y^*)|-D|y|^{\\epsilon}\\def\\part{\\partial_t}\\big)^{p'\/n}\\le C s^{-p'\/n}$$\nand if  $B(0,\\delta)\\subseteq \\Omega$ then pick $s$ so large that $C s^{-p'\/n}<\\delta$. Let $\\epsilon}\\def\\part{\\partial_t<n\/p'$ and  \n$$E_s=\\{y^*\\in S^{n-1}:\\,|g(0,y^*)|>D|g(0,y^*)|^{\\epsilon}\\def\\part{\\partial_t p'\/n}s^{-\\epsilon}\\def\\part{\\partial_t p'\/n}\\}=\\{y^*\\in S^{n-1}:\\,|g(0,y^*)|>D^{n\\over n-\\epsilon}\\def\\part{\\partial_t p'}s^{-{\\epsilon}\\def\\part{\\partial_t p'\\over n-\\epsilon}\\def\\part{\\partial_t p'}}\\}.$$\nThen,\n$$\\eqalign{m_0(s)&\\ge {s^{-p'}\\over n}\\int_{E_s}\\big(|g(0,y^*)|-D|g(0,y^*)|^{\\epsilon}\\def\\part{\\partial_t p'\/n}s^{-\\epsilon}\\def\\part{\\partial_t p'\/n}\\big)^{p'}dy^*\\ge {s^{-p'}\\over n}\\int_{E_s}\\big(|g(0,y^*)|^{p'} -Cs^{-\\sigma}\\big)dy^*\\cr&\n\\ge {s^{-p'}\\over n}\\int_{S^{n-1}}|g(0,y^*)|^{p'}dy^* -Cs^{-p'-\\sigma}\\cr} $$\nwhich means that we have equality in (76).\nFinally, (77) is a simple consequence of (73) and the boundedness of $g$.$$\\eqno\/\\!\/\\!\/$$\n\\vskip-1em\\eject\n\\pf Proof of Theorem 8. The previous Lemma implies that \n$$K_1^*(t)\\le At^{-1\/p'}\\big(1+O(t^{\\epsilon}\\def\\part{\\partial_t})\\big),\\qquad K_2^*(t)\\le Ct^{-1\/p'}\\eqno(78)$$\nso that the exponential inequality (74) follows form Theorem 1.\n\nTo prove sharpness, we appeal to Theorem 4. If the sup in  (75) is attained in $\\Omega$, say WLOG at $x=0$,  then we have equality in the first estimate of (78). Choose $x_m=0$, and let $C_0,\\, m,\\, R$ large enough so that \n$$ \\{y\\in\\Omega: \\,|K(0,y)|>m\\}\\subseteq B(0,C_0m^{-p'\/n})\\subseteq \\Omega\\subseteq B(0,R).$$\nChoosing \n$$r_m=C_0m^{-p'\/n},\\;\\; E_m=B(0,r_m),\\;\\; B_m=B\\big(0,\\ts{1\\over2} r_m\\big)$$\nwe have that conditions a), b), c) of Theorem 4 are satisfies, so all we need to check is \n$$\\int_{\\Omega\\setminus E_m} |K(x,y)-K(0,y)|\\, |K(0,y)|^{p'-1} dy\\le C\\,,\\qquad \\forall x\\in B_m.\\eqno(79)$$\nIt is enough to verify this for $K(x,y)=g(x,(y-x)^*)|x-y|^{ d-n}$. By adding and subtracting $g\\big(x,(y-x)^*\\big)|y|^{ d-n}$ we see that  it suffices to verify\n$$\\int_{r_m\\le |y|\\le R} \\big||x-y|^{ d-n}-|y|^{ d-n}\\big|\\,|y|^{- d}dy\\le C,\\qquad |x|\\le {r_m\\over2}\\eqno(80)$$\nwhich is the same as (62), and\n$$\\int_{r_m\\le |y|\\le R} |g\\big(x,(y-x)^*\\big)-g(0,y^*)|\\,|y|^{-n}dy\\le C,\\qquad |x|\\le {r_m\\over2}.\\eqno(81)$$\nThe H\\\"older continuity hypothesis on $g$ imply \n$$|g\\big(x,(y-x)^*\\big)-g(0,y^*)|\\le C|x|^{\\sigma}+C\\bigg|{y-x\\over|y-x|}-{y\\over|y|}\\bigg|^{\\sigma}\\le C|x|^{\\sigma\/2}|x-y|^{-\\sigma\/2}$$\nbut if $|x|\\le r_m\/2 $ and $|y|\\ge r_m$, then $|x-y|\\ge |y|\/2$\nand we are reduced to \n$$\\int_{r_m\\le |y|\\le R} |x|^{\\sigma\/2}|y|^{-n-\\sigma\/2}dy\\le C,\\qquad |x|\\le {r_m\\over2}$$\nwhich is clearly true.$$\\eqno\/\\!\/\\!\/$$\n\n\\bigskip\\eject\n\\noindent{\\it Sharp inequalities for  general elliptic operators}\\medskip\nWith Theorem 8 at our disposal we are now in a position to extend Adams inequality (2) to rather general elliptic differential operators  of order \n$d<n$. In order for this machinery  to work, we need to make sure that we can write $u=T(Pu)$, for $u$ in a suitable class of smooth functions with compact support where $Pu\\neq0$, and where $T$ is an integral operator with a kernel essentially equal to the  kernel of the parametrix $G$ of $P$. Note that we cannot simply take $T=G$, since $G$ is not, in general, an exact left inverse of P.\n\nSo let $P$ be a general  elliptic differential operator of  order $d$, written as \n$$P=\\sum_{|\\alpha|\\le d} a_\\alpha\\partial^\\alpha$$\non the space of distributions $\\D'(U)$, some open set $U$. The coefficients $a_\\alpha$ are assumed to be $C^\\infty$, complex-valued, and \nthe principal symbol of $P$ then satisfies $p_d(x,\\xi)=p_d^0(x,\\xi)=(2\\pi i)^d\\sum_{|\\alpha|=d} a_\\alpha(x)\\xi^\\alpha\\neq0$, for $\\xi\\neq0$. The adjoint of $P$ is the operator\n$P^*=\\sum_{|\\alpha|\\le d} (-1)^{|\\alpha|}\\partial^\\alpha\\overline{a}_\\alpha.$\n\\smallskip\\medskip\n \n\\proclaim Theorem 10. Let $P$ be an elliptic,  differential operator of order $d<n$ on an open set $U$, with principal symbol $p_d(x,\\xi)$. Let $\\Omega$ be open and bounded with $\\overline\\Omega\\subseteq U$, and let $p=\\displaystyle{n\\over d}$, $\\displaystyle{1\\over p}+\\displaystyle{1\\over p'}=1$. If $P$ is injective  on $C_c^\\infty(\\overline\\Omega)$, then there exists a constant $C$ such that \n$$\\int_{\\Omega} \\exp\\bigg[A^{-1}\\bigg({|u(x)|\\over \\|Pu\\|_p}\\bigg)^{p'}\\bigg]\\,dx\\le C\\eqno(82)$$\nfor all $u\\in W_0^{d,p}(\\Omega)$, with \n$$A={1\\over n}\\,\\sup_{x\\in \\Omega}\\int_{S^{n-1}}|g(x,\\omega)|^{p'}d\\omega\\eqno(83)$$ \n$$g(x,\\omega)=\\bigg({1\\over p_d^0(x,\\cdot)}\\bigg)^\\wedge (\\omega),\\qquad\\omega\\in S^{n-1}.\\eqno(84)$$\nIn the special case  $p=p'=2$, i.e. $d=\\displaystyle{n\\over2}$,  we have \n$$A={1\\over n}\\sup_{x\\in\\Omega} \\int_{S^{n-1}}{1\\over \\big|p_{n\/2}^0(x,\\omega)\\big|^2}d\\omega=\\sup_{x\\in \\Omega}\\int_{\\sR^n}}\\def\\bSn{{\\bS}^n e^{-|p_{n\/2}^0(x,\\xi)|^2}d\\xi.\\eqno(85)$$\n If the supremum in (83) is attained in $\\Omega$, and if the adjoint $P^*$ is injective on $C_c^\\infty(\\overline\\Omega)$ (in particular if $P$ is self-adjoint) then the constant $A^{-1}$ in (82) is sharp.\n\\par \n\\def{\\cal F}{{\\cal F}}\n\\pf Proof. It is enough to assume $u\\in C_c^\\infty(\\Omega)$.  The given hypothesis assure the existence a (properly supported) {$\\Psi$DO}  $T$ of order $-d$ such that $TP u=u$ for any $u\\in C_c^\\infty(\\Omega)$ (in fact for any distribution $u$  with support in $\\overline\\Omega$) (see for example [Ca], Thm 24 and Thm 29). If $G$ denotes the parametrix of $P$ (in $U$) then $T=G+R$, some smoothing operator $R$, and therefore\n$$u(x)=\\int_\\Omega K(x,y)P u(y)dy$$\nwith $K(x,y)$ having an expansion as in (67); in particular\n$$K(x,y)=g\\Big(x,{y-x\\over|y-x|}\\Big)\\,|x-y|^{ d-n}+O\\big(|x-y|^{ d-n+\\epsilon}\\def\\part{\\partial_t}\\big)\\qquad x,y\\in \\Omega$$ \nwith \n$g(x,\\omega)$ as in (84). Clearly this is precisely what one needs in order to apply Theorem 8, and hence prove   \n inequality (82). For the sharpness statement, if $P^*$ is injective on $C_c^\\infty(\\overline\\Omega)$ then $P(Tf)=f$ in $\\Omega$, for any $f\\in \\D'(U)$ (see [Ca], Thm 24),  $P:W_0^{d,p}(\\Omega)\\to L^p(\\Omega)$ has  left inverse $T$, and $T:L^p\\to W^{d,p}(\\Omega)$. Suppose WLOG that the supremum in (83) is attained at $x_0=0\\in\\Omega$ and $B(0,1)\\subset\\subset\\Omega$. If $\\Phi_m$ is the sequence in the sharpness statement of Theorem 8, let $u_m=\\psi T\\Phi_m$, \nwith $\\psi\\in C_c^\\infty(\\Omega)$ and $\\psi=1$ in $B(0,1)$. Then $u_m\\in W^{d,p}_0(\\Omega)$ and it's easy to check using the Leibniz rule that $P u_m=\\psi \\Phi_m+ S \\Phi_m$, where $S$ is a pseudodifferential operator of order  at most $-1$. Hence, since $|\\Phi_m(y)|\\le C |y|^{-d}$ on $\\Omega$, we have $|S\\Phi_m(y)|\\le C|y|^{-d+1}$ and a straightforward estimate shows\n$$\\|Pu_m\\|_p=\\|\\Phi_m\\|_p+O(1),$$\n so that  \n$$\\int_{\\Omega} \\exp\\bigg[\\alpha\\bigg({|u_m(x)|\\over \\|Pu_m\\|_p}\\bigg)^{p'}\\bigg]\\,dx\\ge \\int_{B(0,1)} \\exp\\bigg[\\alpha\\bigg({|T \\Phi_m(x)|\\over \\|\\Phi_m\\|_p+C}\\bigg)^{p'}\\bigg]\\,dx\\to\\infty$$\nfor any $\\alpha>A^{-1}$.\nFinally, when $p=2$ the first formula for $A$ given in (85) is a consequence of  the following  spherical Parseval's formula:     if $f,g\\in C^\\infty(S^{n-1})$ and $E_{-d} (f),\\, E_{d-n} (g)$ are their homogeneous  extensions to ${\\hbox{\\mathbold\\char82}}^n\\setminus 0$ of order $-d$ and $d-n$ respectively $(0<d<n)$, then \n$$\\int_{S^{n-1}}E_{-d}^\\wedge(f)\\overline{E_{d-n}^\\wedge(g)}=\\int_{S^{n-1}} f\\overline g.\\eqno(86)$$\nThe formula was originally found by Koldobsky  in case $f$ and $g$ are powers of Minkowsky functionals of smooth star bodies (see [K] for a proof). In [Mil] the above version is proven for $f,g$ real-valued and even, but a small modification of the proof yields the more general result. The second identity in (85) is obtained by a polar coordinate change. $$\\eqno\/\\!\/\\!\/$$\n\\def{\\cal U}{{\\cal U}}\n\\vskip-2.5em\n\\noindent{\\bf Remarks.}\\smallskip\n \\noindent{\\bf 1.} It is possible to extend slightly Theorem 10 to the case when $P$ does not have a trivial nullspace. Indeed, in the setup of Theorem 10, if $\\ker P$ denotes the nullspace of $P$ among distributions which are supported in $\\overline\\Omega$, then , then $\\ker(P)$ consists of $C^\\infty$ functions, and it is finite-dimensional. \n\\smallskip\\noindent{\\bf 2.} The argument for the sharpness statement above can clearly be used to settle the sharpness statement of Adams  original result, or that of  our Theorem 6, in the case of $\\Delta^{d\/2}$ for $d$ even.\\smallskip\n\\noindent{\\bf 3.} The purpose of the second identity in (85) is that in many situations the exponential integral can actually be evaluated explicitly. See a related calculation in Corollary 14 below.\\smallskip\n\\smallskip \\noindent{\\bf 4.} Both Theorems 8 and 10 can be formulated under a Borel measure satisfying  condition (52), in the same spirit as in Theorems 6 and 7 - the changes are minimal. We kept the usual Euclidean measures to better emphasize the relations between the sharp constants and the operators.\n\n\\smallskip\n\nWe now specialize Theorem 10 to the second order case. Let us start with an elliptic operator \n$$P=\\sum_{j,k=1}^n a_{jk} \\partial_{jk}^2+\\sum_{j=1}^n b_j \\partial_j+ c\\eqno(87)$$\nwith $a_{jk},b_j,c\\in C^\\infty(U)$, {\\it real-valued}, $a_{jk}=a_{kj}$, and let $\\A_x=\\big(a_{jk}(x)\\big)$, an $n\\times n$ symmetric matrix, which we assume to be positive definite. As before we assume that $\\Omega$ is bounded, open and\n $\\overline\\Omega\\subseteq U$. \\medskip\n \n\n\\proclaim Corollary 11. Suppose that $P$ is an elliptic operator as in (87), and injective  on $C_c^\\infty(\\overline \\Omega)$. Then for $n>2$ there exists $C>0$ such that \n$$\\int_\\Omega \\exp\\bigg[n(n-2)^{n\\over n-2}\\omega_{n-1}^{2\\over n-2}\\,\\inf_{x\\in \\Omega}(\\det\\A_x)^{{1\\over n-2}}\\bigg({|u(x)|\\over \\|Pu\\|_{n\/2}}\\bigg)^{n\\over n-2}\\bigg]dx\\le C\\eqno(88)$$\nfor all $u\\in W_0^{2,n\/2}(\\Omega)$.\nIf $\\,\\displaystyle{\\inf_{x\\in \\Omega}}\\det\\A_x$ is attained in $\\Omega$ then the exponential constant in  (88) is sharp.\\par\n\\medskip\n\\noindent{\\bf Note.} If $P$ is strongly  elliptic in $U$, the classical theory (e.g. [GT], Thm 8.9) guarantees that $P$ is certainly injective on $C_c^\\infty(U)$ if $c\\le 0$. \\medskip\n\\pf Proof. All we need to do is apply Theorem 10 to the operator $P$, with \n$$g(x,\\omega)=-{1\\over (2\\pi)^2} \\int_{\\sR^n}}\\def\\bSn{{\\bS}^n {e^{-2\\pi i \\omega\\cdot\\xi}\\over \\xi^T\\A_x\\xi} \\,d\\xi$$\nwhere $\\xi^T$ denotes the transpose of the vector $\\xi$ seen as a column vector. If $\\lambda_1(x),...,\\lambda_n(x)$ denote the positive eigenvalues of $\\A_x$ and if ${\\xi\\over\\sqrt\\lambda}=\\Big({\\xi_1\\over\\sqrt\\lambda_1},...,{\\xi_n\\over\\sqrt\\lambda_n}\\Big)$,  then for some orthogonal matrix $R$\n$$g(x,\\omega)=-{1\\over\\sqrt {\\det \\A_x}}\\int_{\\sR^n}}\\def\\bSn{{\\bS}^n {e^{-2\\pi i R\\omega\\cdot{\\xi\\over\\sqrt\\lambda}}\\over(2\\pi)^2|\\xi|^2} \\,d\\xi=-{c_2\\over \\sqrt {\\det \\A_x}}\\,\\bigg|{R\\omega\\over\\sqrt \\lambda}\\bigg|^{2-n}$$\nwhere $c_2=\\displaystyle{1\\over (n-2)\\omega_{n-1}}$ is the constant in the Newtonian potential, as in (54).\n\\smallskip\nNext, we compute \n$$\\int_{S^{n-1}} |g(x,\\omega)|^{n\\over n-2}d\\omega=\\bigg({c_2\\over\\sqrt {\\det \\A_x}}\\bigg)^{n\\over n-2} \\int_{S^{n-1}}\\bigg|{\\omega\\over\\sqrt \\lambda}\\bigg|^{-n}d\\omega.$$\nBut the computations of the volume (with $x=x^*|x|$)\n$$\\eqalign{{\\omega_{n-1}\\over n}\\,\\sqrt {\\det \\A_x}&=\\Big|\\Big\\{x:\\,\\Big|{x\\over\\sqrt\\lambda}\\Big|<1\\Big\\}\\Big|=\\Big|\\Big\\{x:\\,|x|<\\Big|{x^*\\over\\sqrt\\lambda}\\Big|^{-1}\\Big\\}\\Big|={1\\over n}\\,\\int_{S^{n-1}}\\bigg|{\\omega\\over\\sqrt \\lambda}\\bigg|^{-n}d\\omega\\cr}$$\ngive that \n$\\int_{S^{n-1}}\\big|{\\omega\/\\sqrt \\lambda}\\big|^{-n}d\\omega=\\omega_{n-1}\\sqrt {\\det \\A_x}$, and this concludes the proof.\\hskip4em \/\/\/\\smallskip\\smallskip\n\n\n\\noindent{\\bf Remarks.}\\smallskip\\noindent \n \\noindent{\\bf 1.} In case $b_1=...b_n=c=0$ the result of Corollary 11 can be derived directly from the known asymptotic expansion of the fundamental solution of $P$, and under even less restrictive smoothness conditions on the coefficients. In the case of $\\lambda$-H\\\"older continuous coefficients, ($0<\\lambda<1$) a classical result (See [Mi], Thm 19, VIII) guarantees that the equation $Pu=0$ has a fundamental solution $K(x,y)$ with an expansion\n$$K(x,y)={c_2\\over \\sqrt {\\det \\A_x}}\\Big((x-y)^T\\A_x^{-1}(x-y)\\Big)^{2-n\\over2}\\big(1+O(|x-y|^{\\lambda})\\big).$$\nThis expansion can also be extended to Dini-continuous coefficients or even under  weaker conditions [MMcO]. \nWith the aid of such expansion the calculation of the distribution function of $K$ is straightforward, and produces the same constant as that of the above corollary. For the sharpness result, one just needs to make sure that estimate d) of Theorem 4 is verified, under milder smoothness conditions on the coefficients (and ultimately of the function $g(x,\\omega)$).\n \\smallskip\n\\noindent{\\bf 2.} In [FFV], Thm. 3.5,  an estimate such as (88) is derived  using a different method,  and for elliptic operators with much more general coefficients; the constant produced there is  $n(n-2)^{n\\over n-2}\\omega_{n-1}^{2\\over n-2}$, under the ellipticity hypothesis $\\xi^T\\A_x\\xi\\ge |\\xi|^2$. In such hypothesis and with smoother coefficients,  it is clear that our constant is in general greater (i.e. better), since $\\det\\A_x\\ge1$. \n\n\\medskip\\noindent {\\it Sharp inequalities for vector-valued operators.}\\medskip\nWe now offer a version  of Theorem 10 for vector-valued differential operators of type\n$${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}=(P_j),\\qquad P_j=\\sum_{|\\alpha|\\le d} a_{j\\alpha} \\partial^\\alpha,\\qquad j=1,2,...,\\ell,\\quad \\ell\\in{\\hbox{\\mathbold\\char78}}}\\def\\S{{\\cal S} \\eqno(89)$$\nwith $a_{j\\alpha}\\in C^\\infty$ and complex-valued, with  sharp statements  in the special case $p=2$, i.e.  $d=n\/2$.\n \nThe goal is clearly to extend Adams' inequality for  the operators $\\nabla\\Delta^{d-1\\over2}$ with $d$ odd, by mimicking the integration by parts that leads to the representation formula (57). For the scalar case one can represent $u$ in terms of $Pu$ essentially in a unique way, if $P$ is elliptic and injective;  in the vector-valued situation, on the other hand,   a question of ``optimal representation\" of $u$ in terms  of ${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X} u$ arises, in order to obtain sharpness. The basic idea is to start with a vector-valued differential operator ${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}$ as above,  and assume that for a given operator  ${\\bf Q}=(Q_j)$ of order $d'$,  the operator $L={\\bf Q}^*\\cdot {\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}$ with order $d+d'\\le n$ is elliptic and injective in $C^\\infty_c$, so that it has an inverse $T$ of order $-d-d'$, and a Schwarz kernel $k(x,y)$. One can therefore write $u=(T{\\bf Q}^*)\\cdot {\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X} u$ and apply Theorem 1' to obtain an Adams inequality, with exponential constant given explicitly in terms of the symbols of ${\\bf Q}$ and ${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}$. Clearly one cannot expect such constant to be sharp, given the dependence on $\\bf Q$. We will not state in full generality such result, and for simplicity we will only deal with the case $\\bf Q={\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}$, since in the special situation  \n$p=2$ i.e. $d=n\/2$ a sharpness result can be easily obtained.\n\n\\def{\\bf Y}{{\\bf Y}}\n\nFor   vectors $\\X=(X_j),\\,{\\bf Y}=(Y_j)$ we let $\\X\\cdot {\\bf Y}=\\sum_{j=1}^\\ell X_j Y_j$, $\\,|\\X|=\\big(\\X\\cdot\\overline \\X\\big)^{1\/2}=\\Big(\\sum_1^\\ell|X_j|^2\\Big)^{1\/2}$.\n\n\\proclaim Theorem 12. Let ${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}=(P_j)$ be an operator as in (89), with $d\\le \\displaystyle {n\\over2}$, defined on $\\D'(U)$, some open set $U$. If $\\Omega$ is open and bounded with $\\overline\\Omega\\subseteq U$ and if  $L=\\sum_1^\\ell P_j^* P_j$\nis elliptic on  $U$  and   injective  on $C_c^\\infty(\\overline\\Omega)$, then there exists a constant $C$ such that, with $p=n\/d$, \n$$\\int_{\\Omega} \\exp\\bigg[A^{-1}\\bigg({|u(x)|\\over \\|{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X} u\\|_p}\\bigg)^{p'}\\bigg]\\,dx\\le C\\eqno(90)$$\nfor all $u\\in W_0^{d,p}(\\Omega)$, with\n $$A={1\\over n}\\,\\sup_{x\\in \\Omega}\\int_{S^{n-1}}|{\\bf g}(x,\\omega)|^{p'}d\\omega$$ \n$${\\bf g}(x,z)=\\big(g_j(x,z)\\big),\\qquad g_j^{}(x,z)=\\Bigg({\\overline p_j^0(x,\\cdot)\\over \\displaystyle{\\sum_{k=1}^\\ell |p_k^0(x,\\cdot)|^2}}\\Bigg)^\\wedge(z),$$ \nwhere $p_j^0(x,\\xi)=(2\\pi i)^{d}\\sum_{|\\alpha|=d}a_{j\\alpha}(x)\\xi^\\alpha$ is the principal symbol of $P_j$.\\smallskip\nIn the case $p=2$ i.e. $d=\\displaystyle{n\\over2}$ we have \n$$A={1\\over n}\\,\\sup_{x\\in \\Omega}\\int_{S^{n-1}}\\bigg(\\sum_{j=1}^\\ell |p_j^0(x,\\omega)|^2\\bigg)^{-1}d\\omega=\\sup_{x\\in \\Omega}\\int_{\\sR^n}}\\def\\bSn{{\\bS}^n \\exp\\bigg(-\\!\\!\\sum_{j=1}^\\ell |p_j^0(x,\\xi)|^2\\bigg)d\\xi\\eqno(91)$$ \n and if the supremum in (91) is attained in $\\Omega$, then the constant $A^{-1}$ in (90) is sharp.\n\\par\n\\medskip\n\\pf Proof. The given hypothesis on $L$ imply, just  as before, that  we can write any $u\\in C_c^\\infty(\\Omega)$ as\n$u=T(Lu)=\\sum_{j} TP_j^*(P_j u)$, for a certain {$\\Psi$DO}  $T$ of order $-2d\\ge -n$, with Schwarz kernel $k(x,y)$ and principal symbol $p(x,\\xi)=\\Big(\\sum_{k=1}^\\ell |p_k^0(x,\\xi)|^2\\Big)^{-1}$. Since now $TP_j^*$ is a {$\\Psi$DO} \nof order $-d$, and with principal symbol $\\overline{p}_j^0(x,\\xi)p(x,\\xi)$, then it has a Schwarz kernel $K_j(x,y)$\nso that $$K_j(x,y)=g_j^{}\\big(x,(y-x)^*\\big)|x-y|^{d-n}+O(|x-y|^{d-n+\\epsilon}\\def\\part{\\partial_t}).$$\nThe inequality in (90)  follows now from Theorem 1', since \\def{\\cal K}}\\def\\A{{\\bf A}}\\def\\D{{\\cal D}}\\def\\G{{\\cal G}}\\def\\RR{{\\cal R}{{\\bf K}} if ${\\cal K}}\\def\\A{{\\bf A}}\\def\\D{{\\cal D}}\\def\\G{{\\cal G}}\\def\\RR{{\\cal R}=(K_j)$ then \n$$u=\\int_\\Omega {\\cal K}}\\def\\A{{\\bf A}}\\def\\D{{\\cal D}}\\def\\G{{\\cal G}}\\def\\RR{{\\cal R}(x,y)\\cdot {\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X} u(y)dy$$\nwith $|{\\cal K}}\\def\\A{{\\bf A}}\\def\\D{{\\cal D}}\\def\\G{{\\cal G}}\\def\\RR{{\\cal R}(x,y)|=\\big|{\\bf g}\\big(x,(y-x)^*\\big)\\big|\\,|x-y|^{d-n}+O(|x-y|^{d-n+\\epsilon}\\def\\part{\\partial_t})$\nand the estimates on its distribution functions follow from  Lemma 9. The formula for $A$ given in (91) is a consequence  of the spherical Parseval formula (86).\n\n To prove sharpness of the constant in (90) in the special case $p=2$,  we proceed as in the  proof of Theorem 6. Let the supremum in (91) be achieved at some $x_0\\in \\Omega$ and WLOG assume $x_0=0$. Note that $K_j(x,\\cdot)=P_j k(x,\\cdot)$, where $k$ is the kernel of $T$, and that $k(0,y)=c\\log{1\\over|y|}+O(1)$, some $c>0$, as per (70); let's say that \n$$c\\log{c_0\\over|y|}\\le k(0,y)\\le c\\log{c_1\\over|y|},\\qquad y\\in \\overline\\Omega$$\nfor some $c_0,c_1>0$. Now, using the same $\\varphi$ as in (65), with $r_m\\to0^+$ to be selected later, define\n$$u_m(y)=\\cases{0 &  for $\\;k(0,y)\\le \\delta$\\cr \\cr\\varphi\\big(k(0,y)\\big) & for $\\;\\delta< k(0,y)\\le 1+\\delta$\\cr  k(0,y) & for $\\;1+\\delta< k(0,y)\\le c \\log\\displaystyle{1\\over r_m}-1-\\delta$\\cr   c\\log\\displaystyle{1\\over r_m}-\\varphi\\Big({c\\log\\displaystyle{1\\over r_m}-k(0,y)\\Big)} & for $c\\log\\displaystyle{1\\over r_m}-1-\\delta<k(0,y)\\le c\\log\\displaystyle{1\\over r_m}-\\delta$\\cr \nc\\log\\displaystyle{1\\over r_m}& for $\\;k(0,y)>c\\log\\displaystyle{1\\over r_m}-\\delta$.\\cr}$$\n Then $u_m=0$ if $|y|>c_1 e^{-\\delta\/c}$, hence  we can choose $\\delta$ so  large that the support of $u_m$ is inside $\\Omega$, which implies that   $u_m\\in W^{n\/2,2}_0(\\Omega)$. Additionally,  $u_m=c\\log{1\\over r_m}$ for $|y|<c_0 r_m e^{-\\delta\/c}$, and $u_m(y)=k(0,y)$ for $c_1r_m e^{1+\\delta\\over c}<|y|< c_0 e^{-{1+\\delta\\over c}}$, for $m$ large enough. So \n$$P_j u_m(y)=\\cases{ K_j(0,y) & for $c_1r_m e^{1+\\delta\\over c}< |y|< c_0 e^{-{1+\\delta\\over c}}$\\cr O(1) |y|^{-n\/2}  \n & for $c_0 r_m e^{-\\delta\/c}< |y|<c_1r_m e^{1+\\delta\\over c}$\\cr O(1) & otherwise\\cr}$$\n(here we used the chain and product rules, combined with $|\\partial_y^\\alpha k(0,y)|\\le C |y|^{-|\\alpha|}$, for $|\\alpha|>0$, since $\\partial^\\alpha k$ is the kernel of the operator $\\partial^\\alpha T$, which has order $|\\alpha|-n$).\n\\eject\nNow  choose $r_m$ so that $\\{y\\in \\Omega : |{\\cal K}}\\def\\A{{\\bf A}}\\def\\D{{\\cal D}}\\def\\G{{\\cal G}}\\def\\RR{{\\cal R}(0,y)|>m\\}\\subseteq B(0,C m^{-2\/n})\\subseteq B(0, c_1 r_m e^{1+\\delta\\over c})$, and therefore, we can apply (45) of Theorem 4 with $E_m=B(0, c_1 r_me^{1+\\delta\\over c})$ to conclude\n$$\\int_{\\Omega\\setminus B(0, c_1 r_me^{(1+\\delta)\/ c})} |{\\cal K}}\\def\\A{{\\bf A}}\\def\\D{{\\cal D}}\\def\\G{{\\cal G}}\\def\\RR{{\\cal R}(0,y)|^2dy=A\\log{1\\over r_m}+O(1)$$\nwhich allows us to conclude $\\|{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X} u_m\\|_2^2=A\\log{1\\over r_m}+O(1)$ \nand the sharpness of the exponential constant follows immediately from (49), just as in the proof of Theorem 6. $$\\eqno\/\\!\/\\!\/$$\n\n\n\n\n\nWe will give one first application of the above theorem to first order operators. Consider a family of  operators \n$${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}=\\big(P_j\\big)_{j=1}^n,\\quad P_j=\\sum_{k=1}^n a_{jk} \\partial_k+b_j\\eqno(92)$$\nwith  $a_{jk},\\, b_j$  real-valued and $C^\\infty$  on some open set $U\\supseteq \\overline\\Omega$, with $\\Omega$ bounded.\n\n\\proclaim Corollary 13. Suppose that $\\A_x=\\big(a_{jk}(x)\\big)$ is invertible on $U$ and that  $L=\\sum_{j=1}^n P_j^*P_j$ is injective  on $C_c^\\infty(\\overline \\Omega)$. Then, for $n>1$ there exists $C>0$ such that \n$$\\int_\\Omega \\exp\\bigg[n\\omega_{n-1}^{1\\over n-1}\\,\\inf_{x\\in \\Omega}|\\det\\A_x|^{{1\\over n-1}}\\bigg({|u(x)|\\over \\|{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X} u\\|_{n}}\\bigg)^{n\\over n-1}\\bigg]dx\\le C\\eqno(93)$$\nfor all $u\\in W_0^{1,n}(\\Omega)$.\nIf  $\\,\\displaystyle{\\inf_{x\\in \\Omega}}|\\det\\A_x|$ is attained in $\\Omega$ then the exponential constant in  (93) is sharp.\\par\n\\bigskip\n\\noindent{\\bf Note.} In [FFV], Theorem 3.3, a similar estimate is given for less regular  coefficients,  under the condition $\\xi^T\\A_x\\xi\\ge |\\xi|^2$, and with exponential  constant $n\\omega_{n-1}^{1\\over n-1}$, which is smaller than the one given in the above Corollary.\\smallskip\n\n\n\n\\medskip\\pf Proof. The proof of (93) is just an application of Theorem 12. One just has to first compute ${\\bf g}$, proceeding like in the proof of Corollary 12: if ${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_0=\\Big(\\sum_j a_{ij}\\partial_j\\Big)=\\A_x\\cdot\\nabla$\n$$\\eqalign{-{\\bf g}(x,z)&=\\bigg({(2\\pi i)\\A_x\\xi\\over(2\\pi)^2 |\\A_x\\xi|^2}\\bigg)^\\wedge(z)= {\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_0\\bigg({1\\over(2\\pi)^2 |\\A_x\\xi|^2}\\bigg)^\\wedge(z)={1\\over |\\det \\A_x|}{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_0\\bigg({1\\over (2\\pi)^2|\\xi|^{2}}\\bigg)^\\wedge\\big((\\A_x^{-1})^Tz\\big)\\cr&={c_2\\over |\\det \\A_x|}{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_0 \\big|(\\A_x^{-1})^T z\\big|^{2-n}={(2-n)c_2\\over |\\det \\A_x|} \\big((\\A_x^{-1})^Tz\\big)\\big|(\\A_x^{-1})^T z\\big|^{-n}\\cr}$$\nsince if $\\A_x^{-1}=(a_{jk}')$ then  \n$\\sum_j a_{ij}\\partial_j\\big|(\\A_x^{-1})^Tz\\big|^{2-n}=(2-n)\\big|(\\A_x^{-1})^Tz\\big|^{-n}\\sum_{j,k} a_{ij}a_{jk}'\\big((\\A_x^{-1})^T z\\big)_k. $ Estimate (93) follows since\n$$A={1\\over n}\\sup_x\\int_{S^{n-1}}|{\\bf g}(x,\\omega)|^{n\\over n-1}d\\omega={1\\over n}\\sup_x{1\\over (\\omega_{n-1}|\\det \\A_x|)^{n\\over n-1}} \\int_{S^{n-1}}\\big|(\\A_x^{-1})^T\\omega\\big|^{-n}d\\omega$$\nand $\\displaystyle\\int_{S^{n-1}}\\big|(\\A_x^{-1})^T\\omega\\big|^{-n}d\\omega=\\omega_{n-1}|\\det \\A_x|$. For the sharpness statement, suppose  WLOG that $\\,\\displaystyle{\\inf_{x\\in \\Omega}}|\\det\\A_x|$ is attained $x_0=0\\in\\Omega$ and that the ellipsoid $\\{y:|\\A_0^{-1}y|<1\\}\\subseteq \\Omega$. Take any $r_m\\downarrow 0$, $r_m<1$, and let\n$$u_m=\\cases{\\log|\\A_0^{-1}y|^{-1} & if $r_m<|\\A_0^{-1}y|<1$\\cr\n\\log r_m^{-1} & if $|\\A_0^{-1}y|\\le r_m$\\cr\n0 & if $|\\A_0^{-1}y|\\ge1.$\\cr}$$\n Then $u_m\\in W^{1,n}(\\Omega)$, ${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X} u_m(y)=-(\\A_0^{-1}y)|\\A_0^{-1}y|^{-2}+O\\big(\\log|\\A_0^{-1}y|^{-1}\\big)$ if $r_m<|\\A_0^{-1}y|<1$, and it's easy to check that \n$\\|{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X} u_m\\|_n^n=\\omega_{n-1} |\\det\\A_0|\\log{1\\over r_m}+O(1)$. The result follows from (49), with $B_m=\\{y: |\\A_0^{-1}y|<r_m\\}$.\n$$\\eqno\/\\!\/\\!\/$$\n\n\nAs a second and final quick application of Theorem 12 we consider in ${\\hbox{\\mathbold\\char82}}^4$ the  second order operators \n$${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_1=\\big(\\partial_{11}^2,\\,\\partial_{22}^2,\\,\\partial_{33}^2,\\,\\partial_{44}^2\\big)\\,\n\\quad{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_2=\\big(\\partial_{11}^2+\\partial_{22}^2,\\,\\partial_{33}^2+\\partial_{44}^2\\big)\\,\\quad \n{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_3=\\big(\\partial_{11}^2+\\partial_{22}^2+\\partial_{33}^2,\\,\\partial_{44}^2\\big)$$ \n\\proclaim Corollary 14. Let $\\Omega\\subseteq {\\hbox{\\mathbold\\char82}}^4$ be open and bounded. Then there exists $C>0$ such that for $j=1,2,3$\n$$\\int_{\\Omega} \\exp\\bigg[\\,B_j\\,\\bigg({|u(x)|\\over\\|{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_j u\\|_2}\\bigg)^2\\bigg] dx\\le C\\eqno(94)$$\nwith\n$$B_1={\\pi^4\\over \\Gamma\\big({5\\over4}\\big)^4},\\qquad B_2=64 \\pi,\\qquad B_3={16\\pi^{5\/2}\\over\\Gamma\\big({3\\over4}\\big)}$$\nfor any $u\\in W_0^{2,2}(\\Omega)$, and the constants $B_j$  are sharp.\\par\n\nNote that the constant $32\\pi^2$ in the sharp inequality \n$$\\int_{\\Omega} \\exp\\bigg[32\\pi^2\\bigg({|u(x)|\\over\\|\\Delta u\\|_2}\\bigg)^2\\bigg] dx\\le C$$\nis bigger than  all of the constants  in (94), in fact $32\\pi^2>B_3>B_2>B_1$; this is   consistent with $\\|\\Delta u\\|_2\\ge \\|{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_3 u\\|_2\\ge \\|{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_2u\\|_2\\ge\\|{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_1u\\|_2$,  which is  easily seen via Fourier transform.\n\\medskip\n\\pf Proof. We can apply Theorem 12, since the operator $L={\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_1^*\\cdot{\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_1=\\sum_1^4 {\\partial^4\\over\\partial x_j^4}$ is elliptic and injective on $C_c^\\infty(\\overline\\Omega)$, and the same is true for ${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_2^*\\cdot {\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_2$ and ${\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_3^*\\cdot {\\bf P}}\\def\\G{{\\bf G}}\\def{\\bf g}{{\\bf g}}\\def\\X{{\\bf X}_3$. The computation of the constants follows easily from (91) and the identity\n$$\\int_{{\\hbox{\\smathbold\\char82}}}\\def\\mR{{\\hbox{\\mmathbold\\char82}}^m} \\exp\\bigg[-\\bigg(\\sum_{j=1}^m x_j^2\\bigg)^{p\/2}\\bigg] dx={2\\pi^{m\/2}\\Gamma\\big(1+{m\\over p}\\big)\\over m\\Gamma\\big({m\\over2}\\big)}$$\nvalid for $m\\in {\\hbox{\\mathbold\\char78}}}\\def\\S{{\\cal S} $ and $p>0$. Note that $B_1^{-1}$ is in fact the volume of the convex body $\\Big\\{x\\in {\\hbox{\\mathbold\\char82}}^4:\\,\\sum_1^4 x_j^4<1\\Big\\}$  (see for example [K]).\n$$\\eqno\/\\!\/\\!\/$$\\eject\n\\bigskip\n\\noindent {\\bf 6.  Sharp Adams inequalities for  sums of weighted potentials.}\n\\medskip\n\\smallskip\n As another   illustration of how   Theorems 1 and 4 can be used, we offer an  extension of Adams' inequality (3) in a different direction:\n\n\n\n\\bigskip\\proclaim Theorem 15.  Let $\\Omega,\\,\\Omega'$ be  bounded domains of $\\R^n}\\def\\B{{\\cal B}$,  $a_1,...,a_N\\in\\R^n}\\def\\B{{\\cal B}$,$\\,a_j\\neq a_k,\\, j\\neq k$. Let ${\\cal U}$ be a bounded domain of ${\\hbox{\\mathbold\\char82}}^n\\times{\\hbox{\\mathbold\\char82}}^n$, with  $\\Omega'\\times\\Omega\\subset\\subset {\\cal U}$, and  let  $g_j:\\overline{\\cal U}\\to{\\hbox{\\mathbold\\char82}}$,  be H\\\"older continuous of order $\\sigma_j\\in(0,1],\\, j=1,2,...N$. For $0< d<n$, $\\,p=\\displaystyle{n\\over d}$, $\\,\\displaystyle{{1\\over p}+{1\\over p'}}=1$, let, for $x\\in \\Omega'$ and $y\\in\\Omega$,\n$$K(x,y)=\\sum_{j=1}^N g_j(x,y)|x+a_j-y|^{ d-n},\\qquad Tf(x)=\\int_\\Omega K(x,y)f(y)dy,\\quad f\\in L^p(\\Omega),$$\nIf ${\\bf g}=(g_1,...g_N)$ and \n$$M({\\bf g}):=\\sup\\bigg\\{\\sum_{j=1}^N |g_j(x,x+a_j)|^{p'},\\;x\\in\\Omega',\\,(x,x+a_j)\\in\\overline{\\cal U},\\,j=1,...N\\bigg\\}>0,\\eqno(95)$$\nthen there exists $C$ such that\nfor any $f\\in L^p(\\Omega)$\n$$\\int_{\\Omega'} \\exp\\bigg[ {n\\over\\omega_{n-1} M({\\bf g})}\\bigg({|Tf|\\over \\|f\\|_{p}}\\bigg)^{p'}\\,\\bigg]dx\\le C\\eqno(96)$$\nwith   $C$ independent of $f$. \nIf \n$$\\Omega^*:=\\Omega'\\cap\\bigcap_{j=1}^N(\\Omega-a_j)\\neq\\emptyset\\eqno(97)$$\nand $M({\\bf g})$ is attained on $\\Omega^*$, then the constant $\\displaystyle{n\\over\\omega_{n-1} M({\\bf g})}$ is sharp in (96), i.e. it cannot be replaced by a larger constant.\n\\par\n\\medskip\n\n\\pf Proof.  Fix $x\\in \\Omega'$. If $\\delta>0$ is such that $\\delta<{\\rm dist} \\big(\\overline{\\Omega'}\\times\\overline\\Omega\\,,\\,\\overline {\\cal U}^c\\big)$, and   $B(a_j,\\delta)\\cap B(a_k,\\delta)=\\emptyset$,for $j\\neq k$, $j,k=1,...N$,  then for $s>s_1:=N\\delta^{ d-n}\\max_j \\|g_j\\|_\\infty^{}$ we have\n\n$$\\big|\\{y\\in\\Omega:\\,|K(x,y)|>s\\,\\}\\big|=\\sum_{j=1}^N\\big|\\{y\\in\\Omega\\cap B(x+a_j,\\delta):\\, \n|K(x,y)|>s\\}\\big|$$\n\nWith our choice of $\\delta$ it's clear that if $(x,x+a_j)\\notin \\overline{\\cal U}$ then $\\Omega\\cap B(x+a_j,\\delta)=\\emptyset$ so \n$$\\big|\\{y\\in\\Omega\\cap B(x+a_j,\\delta):\\, |K(x,y)|>s\\}\\big|=0$$\nfor any $s>s_1$ (in fact for  any $s>0$).\n\n\nAssume that  $(x,x+a_j)\\in\\overline{\\cal U}$ and $y\\in \\Omega\\cap B(x+a_j,\\delta)$. Then \n\n$$|K(x,y)|\\le |g_j(x,y)|\\,|x+a_j-y|^{ d-n}+C\\delta^{ d-n}\\le |g_j(x,x+a_j)|\\,|x+a_j-y|^{ d-n}+C|x+a_j-y|^{ d-n+\\epsilon}$$\nsome $\\epsilon>0$, $\\epsilon<n- d$ and $C$ independent on $x,y$. As a consequence, if $|K(x,y)|>s$ then\n$$|x+a_j-y|<s^{-1\/(n- d)}\\big(|g_j(x,x+a_j)|+C|x+a_j-y|^\\epsilon\\big)^{1\/(n- d)}\\le Cs^{-1\/(n- d)}$$\n\nand \n\n$$\\eqalignno{\\big|\\{y\\in\\Omega\\cap B(x+a_j,\\delta):\\, \n|K(x,y)|>s\\}\\big|&\\le {\\omega_{n-1}\\over n}\\, s^{-n\/(n- d)}\\big(|g_j(x,x+a_j)|+Cs^{-\\epsilon\/(n- d)}\\big)^{n\/(n- d)}\\cr&\\le{\\omega_{n-1}\\over n}\\, s^{-p'}|g_j(x,x+a_j)|^{p'}+Cs^{-p'-\\sigma}&(98)\\cr} $$\n\nsome $\\sigma>0$ (we used here, for example, that $|(a+b)^\\nu-b^\\nu|\\le Ca^{\\min\\{1,\\nu\\}}$ if $\\nu>0$ and $a,b\\in[0,K]$, some fixed $K>0$, C independent of $a,b$).\n\nNow we see that if $x\\in \\Omega'$  and $(x,x+a_j)\\in\\overline{\\cal U}$ for all $j$, then\n$$\\big|\\{y\\in\\Omega:\\,|K(x,y)|>s\\,\\}\\big|\\le s^{-p'}{\\omega_{n-1}\\over n}\\,\\sum_{j=1}^N |g_j(x,x+a_j)|^{p'}+O\\big(s^{-p'-\\sigma}\\big),\\quad \\forall s> s_1\\eqno(99)$$\n(with $|O\\big(s^{-p'-\\sigma}\\big)|\\le Cs^{-p'-\\sigma}$, $C$ independent of $x,s$), from which it follows that \n\n$$\\sup_{x\\in\\Omega'} m\\big(K(x,\\cdot),s\\big)\\le s^{-p'}M({\\bf g})+O\\big(s^{-p'-\\sigma}\\big).\\eqno(100)\n$$\n\nOn the other hand, the same argument used to derive (99) can be used to show\n\n$$\\big|\\{x\\in\\Omega':\\,|K(x,y)|>s\\,\\}\\big|\\le B s^{-p'}\\eqno(101)$$\nfor all $s>s_1$, and $y\\in\\Omega$, for some $B>0$ independent of $y$.  \n\nEstimate (96) now follows from Theorem 1, using (100), (101) together with Fact 3.\n\n\n\nNow assume that $\\Omega^*\\neq\\emptyset$ and that the sup in (95) is attained inside $\\Omega^*$, say at $x^*$. WLOG we can assume that $x^*=0$ (indeed it's enough to perform a translation by $x^*$ in both the $x$ and the $y$ variables). If $y\\in \\Omega\\cap B(x+a_j,\\delta)$\n$$|K(0,y)|\\ge |g_j(0,a_j)|\\,|a_j-y|^{ d-n}-C|a_j-y|^{ d-n+\\epsilon}\\def\\part{\\partial_t}$$\nand $|g_j(0,a_j)|\\,|a_j-y|^{ d-n}-C|a_j-y|^{ d-n+\\epsilon}\\def\\part{\\partial_t}>s$ if and only if\n$$|a_j-y|<s^{-1\/(n- d)}\\big(|g_j(0,a_j)|-C|a_j-y|^\\epsilon\\big)^{1\/(n- d)}.\n$$\n\nThen letting \n\n$$\\phi(s):=s^{-1\/(n- d)}\\big(|g_j(0,a_j)|-Cs^{-\\epsilon}\\def\\part{\\partial_t\/(n- d)}|g_j(0,a_j)|^{\\epsilon}\\def\\part{\\partial_t\/(n- d)}\\big)^{1\/(n- d)}\\eqno(102)$$ we have \n$$\\eqalign{&\\big\\{y\\in\\Omega\\cap B(a_j,\\delta):\\,|a_j-y|<\\phi(s)\\big\\}\\cr&\\subseteq \\big\\{y\\in\\Omega\\cap B(a_j,\\delta):\\,|a_j-y|<s^{-1\/(n- d)}\\big(|g_j(0,a_j)|-C|a_j-y|^\\epsilon\\big)^{1\/(n- d)}\\big\\}\\cr\n&\\subseteq\\big\\{y\\in\\Omega\\cap B(a_j,\\delta):\\,|K(0,y)|>s\\big\\}.\\cr}\\eqno(103)$$\n\nSince $a_j\\in\\Omega$ let $\\delta_0>0$ be such that $B(a_j,\\delta_0)\\subseteq\\Omega$. There exists $s_0>s_1$ such that $0\\le \\phi(s)<\\delta_0$ for all $s\\ge s_0$, so that \n\n$$\\big|\\big\\{y\\in\\Omega\\cap B(a_j,\\delta):\\,|K(0,y)|>s\\big\\}\\big|\\ge {\\omega_{n-1}\\over n}\\,\\big(\\phi(s)\\big)^n\\ge{\\omega_{n-1}\\over n}\\, s^{-p'}|g_j(0,a_j)|^{p'}-Cs^{-p'-\\sigma}$$\nfor all $s\\ge s_0$.\n\nThis means that for all $s>s_0$\n$$\\big|\\{y\\in\\Omega:\\,|K(0,y)|>s\\,\\}\\big|=s^{-p'}{\\omega_{n-1}\\over n}\\,\\sum_{j=1}^N |g_j(0,a_j)|^{p'}+O\\big(s^{-p'-\\sigma}\\big)=s^{-p'}M({\\bf g})+O\\big(s^{-p'-\\sigma}\\big).$$\n\n\\smallskip\\noindent\n\nNow let us choose $x_m=0$ for $m\\in{\\hbox{\\mathbold\\char78}}}\\def\\S{{\\cal S} $, \n\n$$E_m=\\{y\\in\\Omega:\\,|K(0,y)|>m\\,\\}=\\bigcup_{j=1}^N\\{y\\in\\Omega\\cap B(a_j,\\delta):\\, \n|K(0,y)|>m\\}$$\nthe union being disjoint for $m>s_1$. From  ii) we have $|E_m|=m^{-p'}M({\\bf g})+O(m^{-p'\n-\\sigma})\\to0$ as $m\\to\\infty$. Moreover, from (102) and (103), if $g(0,a_j)\\neq0$ then $\\{y\\in\\Omega\\cap B(a_j,\\delta):\\, \n|K(0,y)|>m\\}$ contains a ball of center $a_j$ and radius $C_j m^{-p'\/n}$ some $C_j>0$, for all $m>m_j>s_1$; let $C_0$ be the smallest of such $C_j$ and let $$r_m=C_0 m^{-p'\/n},\\quad B_m=B\\big(0,\\ts{1\\over2}r_m\\big).\\eqno(104)$$ \n\n\nWith these choices conditions a), b), c)  of Theorem 4 are satisfied, so all we need is to check (43), i.e.\n$$\\int_{\\Omega\\setminus E_m} |K(x,y)-K(0,y)|\\, |K(0,y)|^{p'-1} dy\\le C\\,,\\qquad \\forall x\\in B_m\\eqno(105)$$\nsome $C$ independent of $x$ and $m$.\n\nNow observe the following elementary inequalities, valid for any $x\\in \\Omega'$ and $y\\in \\Omega$\n$$\\eqalign{|K(0,y)|^{p'-1}&\\le C\\sum_{j=1}^N |g_j(0,y)|^{ d\/(n- d)}|y-a_j|^{- d}\\cr&\\le C\\sum_{j=1}^N\\big(|y-a_j|^{\\epsilon}\\def\\part{\\partial_t d\/(n- d)}+|g(0,a_j)|^{ d\/(n- d)}\\big)|y-a_j|^{- d}\\cr}\\eqno(106)$$\n$$\\eqalign{\\big|g_j(x,y)|x+a_j-y|^{ d-n}-g_j&(0,y)|a_j-y|^{ d-n}\\big|\\le C\\big(|y-a_j|^\\epsilon}\\def\\part{\\partial_t+|x|^\\epsilon}\\def\\part{\\partial_t\\big)|x+a_j-y|^{ d-n}\\cr&\\qquad +|g_j(0,a_j)|\\big||x+a_j-y|^{ d-n}-|a_j-y|^{ d-n}\\big|\\cr}\\eqno(107)\n$$\n\n$$\\eqalign{|K(x,&y)-K(0,y)|\\, |K(0,y)|^{p'-1}\\le C\\sum_{j=1}^N\\bigg\\{|x|^\\epsilon}\\def\\part{\\partial_t|y-a_j|^{- d}|x+a_j-y|^{ d-n}+\\cr& +|g(0,a_j)|^{n\/(n- d)}|y-a_j|^{- d}\\big||x+a_j-y|^{ d-n}-|a_j-y|^{ d-n}\\big|\\bigg\\}+\\Phi(x,y)\\cr}\\eqno(108)$$\nwhere $\\Phi(x,y)\\ge0 $ is integrable in $y\\in B(0,R)$ some $R$ large enough so that $\\int_{B(0,R)}\\Phi(x,y)dy\\le C$, independent of   $x\\in \\Omega'$. \n\n\nBy virtue of (108) it is enough to consider those $j$ for which $g(0,a_j)\\neq0$, and for such $j$ we can write $M\\setminus\\subseteq M\\setminus B(a_j,r_m)$ (recall the definition of $r_m$ in (104)). Thus, it all boils down to (62), which we already checked, and the estimate\n$$\\sup_{|x|\\le r_m\/2}\\;\\int_{r_m\\le|y|\\le R} |x|^\\epsilon}\\def\\part{\\partial_t|y|^{- d}|x-y|^{ d-n}dy\\le C,\\eqno(109)$$\nwhich is an easy consequence of (63).\n\n$$\\eqno\/\\!\/\\!\/$$\n\n\\bigskip\n\n\\noindent{\\bf Remarks.} \\smallskip\\item{\\bf 1.} From the proof above it should be  apparent that Theorem 15 holds verbatim for kernels of type\n$$K(x,y)=\\sum_{j=1}^N g_j(x,y)|x+a_j-y|^{ d-n}\\bigg[1+O\\bigg(\\sum_{j=1}^N |x+a_j-y|^{\\epsilon}\\def\\part{\\partial_t_j}\\bigg)\\bigg]\\eqno(110)$$\nwhere $\\epsilon}\\def\\part{\\partial_t_1,...\\epsilon}\\def\\part{\\partial_t_N>0$.\n\\smallskip\n\\item{\\bf 2.} The regularity hypothesis on the $g_j$ can be somewhat relaxed to an integral\ncondition of type (43).\\smallskip  \n\\item{\\bf 3.} If the sup defining $M({\\bf g})$ is not attained in $\\Omega^*$, or if $\\Omega^*$ is empty, then\nthe sharp constant in (96) will in general be larger, and the geometries of  the domains could play a definite role. For example, if $K(x,y)=|x-y|^{ d-n}$, and  $\\Omega',\\, \\Omega$ are two open balls with empty intersection but tangent to one another (or two $C^1$ domains with the same property), then\n$M({\\bf g})=1$, but it's easy to see that the sharp constant in (96) is $2n\/\\omega_{n-1}$. This can bee seen by explicit asymptotics of the distribution function of the kernel with the given domains, together with Theorems 1 and 4. Similar considerations could be made if $\\partial \\Omega$ has corners, or even positive measure.\nOn the other hand, if $K(x,y)=|x+e_1-y|^{ d-n}+|x-e_1-y|^{ d-n}$, with $e_1=(1,0,..,0)$, and $\\Omega'=B(0,10),\\, \\Omega=B\\big(0,{1\\over2}\\big)$, then $\\Omega^*=\\emptyset$, $M({\\bf g})=2$, but the sharp constant in (96) is $n\/\\omega_{n-1}$. This can be seen for example by splitting $\\Omega'$ into two halves  each containing  $e_1$ or $-e_1$, and  noticing that in each half ony one of the two potentials is really effective (i.e. theorems 1 and 4 apply in each half separately).\n\\bigskip\nOn the $n-$dimensional Euclidean sphere $S^n$  Theorem 15 takes a somewhat  simpler form. Let $\\eta,\\xi$ denote points on $S^n$, and let $d\\eta$ denote the standard volume element of $S^n$. \n\n\\proclaim Theorem 16. On $S^n$ consider an operator \n$$Tf(\\xi)=\\int_{S^n}}\\def\\wtilde{\\widetilde K(\\xi,\\eta)f(\\eta)d\\eta,\\qquad f\\in L^1(S^n)$$\nwith $$K(\\xi,\\eta)=\\sum_{j=1}^N g_j(\\xi,\\eta)|R_j\\xi-\\eta|^{ d-n}+O\\Big(\\sum_{j=1}^N |R_j\\xi-\\eta|^{ d-n+\\epsilon_j}\\Big),\\quad 0< d<n,\\;\\sigma_j>0$$\nfor some $R_1,...,R_N\\in SO(n)$, and $g_j:S^n\\times S^n\\to{\\hbox{\\mathbold\\char82}}$ H\\\"older continuous of orders $\\sigma_1,...,\\sigma_N\\in(0,1]$. If $p=\\displaystyle{n\\over d},\\,\\displaystyle{{1\\over p}+{1\\over p'}=1}$, $\\,{\\bf g}=(g_1,...,g_N^{})$ and if\n$$M({\\bf g})=\\max_{\\xi\\in S^n}\\sum_{j=1}^N |g_j(\\xi,R_j\\xi)|^{p'}>0,$$\n then there exists $C$ so that\n$$\\int_{S^n}}\\def\\wtilde{\\widetilde \\exp\\bigg[{n\\over \\omega_n M({\\bf g})}\\bigg({|Tf|\\over\\|f\\|_p}\\bigg)^{p'}\\bigg]\\,d\\xi\\le C\\eqno(111)$$\nfor any $f\\in L^p(S^n)$. The constant $\\displaystyle{n\\over \\omega_n M({\\bf g})}$ in (111) is sharp.\n\\par\nThe proof of this theorem is identical to the one of Theorem 15, with the obvious modifications, and with the additional simplifications due the the compactness of $S^n$.\\bigskip\n\nOn a compact Riemannian manifold $M$, with volume element $dV(P)$  and geodesic distance $d(P,Q)$, we have the following slight extension of Fontana's result ([F], Thm. 1.9):\n\n\\proclaim Theorem 17. On the compact Riemannian manifold $M$ consider an integral operator \n$$Tf(P)=\\int_M K(P,Q)f(Q)dV(Q),\\qquad f\\in L^1(M)$$\nwith\n$$K(P,Q)=g(P,Q)\\,d(P,Q)^{ d-n}+O\\big(d(P,Q)^{ d-n+\\epsilon}\\def\\part{\\partial_t}\\big),\\qquad 0< d<n,\\;\\epsilon}\\def\\part{\\partial_t>0$$\nwith $g:M\\times M\\to{\\hbox{\\mathbold\\char82}}$ H\\\"older continuous of order $\\sigma\\in(0,1].$ \nIf $p=\\displaystyle{n\\over d},\\,\\displaystyle{{1\\over p}+{1\\over p'}=1}$ and if\n$$M(g)=\\max_{P\\in M}|g(P,P)|^{p'}>0,$$\n then there exists $C$ so that\n$$\\int_M \\exp\\bigg[{n\\over \\omega_n M(g)}\\bigg({|Tf|\\over\\|f\\|_p}\\bigg)^{p'}\\bigg]\\,dV(P)\\le C\\eqno(112)$$\nfor any $f\\in L^p(M)$. The constant $\\displaystyle{n\\over \\omega_n M(g)}$ in (112) is sharp.\n\\par\nThe proof of Theorem 17 is a consequence of Theorems 1 and 4, and a sharp asymptotic estimate\nof the distribution function of $K$, which is the same one as in the Euclidean case (Thm 15)\ngiven the fact that that the volume of a small geodesic ball is asymptotically the same as that of a Euclidean ball.\n\\bigskip\n\\noindent{\\bf 7. Sharp Adams inequalities on the CR sphere}\\bigskip\nAs we mentioned in the introduction, Moser-Trudinger inequalities have recently been introduced in the context of CR-manifolds, first by Cohn and Lu [CL1,2] and more recently by Branson, Fontana, Morpurgo [BFM].  In [BFM], a special case of Theorem 1 of the present paper was quoted and used to derive sharp Adams inequalities for a class of convolution operators on the CR sphere ([BFM], Thm. 2.2.). The proof that such inequalities are sharp was only hinted in [BFM]; in this section we will provide a more detailed argument as an application of Theorem 4. \n\nWe will now briefly recall \nthe main setup.\nLet $\\Sn$ be the $(2n+1)-$dimensional sphere with its standard CR structure, i.e. that induced naturally from the ambient space ${\\hbox{\\mathbold\\char67}}}\\def \\sC{{\\hbox{\\smathbold\\char67}}^{n+1}$, endowed with Hermitian product $\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}$, where $\\zeta=(\\zeta_1,...,\\zn)$, $\\eta=(\\eta_1,...,\\eta_{n+1})$. The homogeneous dimension of $\\Sn$ is denoted by $Q=2n+2$. Let $d\\zeta$ be the standard volume element of the sphere, and $\\omega_{2n+1}=2\\pi^{n+1}\/n!$ its volume; the average of a function $F$ on $\\Sn$ is denoted by $\\displaystyle-\\hskip-1.1em\\int F$. \n\nThe Heisenberg group $\\Hn$, with elements $(z,t)\\in {\\hbox{\\mathbold\\char67}}}\\def \\sC{{\\hbox{\\smathbold\\char67}}^n\\times{\\hbox{\\mathbold\\char82}}$ and group law\n$(z,t)(z',t')=(z+z',t+t'+2\\Im z\\cdot \\overline z')$\nis biholomophically equivalent to $\\Sn$ via the Cayley transform ${\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}:\\Hn\\to S^{2n+1}\\setminus(0,0,...,0,-1)$  given by \n$${\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}(z,t)=\\Big({2z\\over 1+|z|^2+i t},{1-|z|^2-i t\\over 1+|z|^2+i t}\\Big)$$\nand with inverse\n$${\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}^{-1}(\\zeta)=\\Big({\\zeta_1\\over1+\\zeta_{n+1}},...,{\\zeta_n\\over1+\\zeta_{n+1}},Im {1-\\zeta_{n+1}\\over\n  1+\\zeta_{n+1}}\\Big).$$ \n\n\nThe homogeneous norm on $\\Hn$ is defined by \n$$|(z,t)|=(|z|^4+t^2)^{1\/4}$$\nand the distance from   $u=(z,t)$ and  $v=(z',t')$ is given as\n$$d((z,t),(z',t')):=|v^{-1}u|=\\big(|z-z'|^4+(t-t'-2\\Im(z\\overline\nz'))^2\\big)^{1\/4}$$\n\nOn the sphere the distance function is defined as \n$$d(\\zeta,\\eta)^2:=2|1-\\zeta\\cdot \\overline\\eta|=\\big|\\,|\\zeta-\\eta|^2-2i\n\\,\\Im(\\zeta\\cdot\\overline\\eta)\\big|=\n\\big(|\\zeta-\\eta|^4+4\\cdot{\\rm \\Im}^2(\\zeta\\cdot\\overline\\eta)\\big)^{1\/2}$$\nand a simple calculation shows that if $u=(z,t),v=(z',t')$, and $\\zeta={\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}(u),\\,\\eta={\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}(v)$.\nthen\n$${|1-\\zeta\\cdot \\overline\\eta|\\over\n  2}=|v^{-1}u|^2\\big((1+|z|^2)^2+t^2\\big)^{-1\/2}\\big((1+|z'|^2)^2+(t')^2\\big)^{-1\/2}\\eqno(113)$$\nFurther, we let \n\n$$u=(z,t)\\in\\Hn,\\quad \\Sigma=\\{u\\in\\Hn:\\,|u|=1\\},\\quad u^*={u\\over|u|}=(z^*,t^*)\\in\\Sigma$$\n$$\\zeta={\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}(u),\\quad {1-\\zn\\over 1+\\zn}=|z|^2+it=(|z|^4+t^2)^{1\/2} e^{i\\theta},\\quad {\\hbox{\\gothic\\char78}}={\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}(0,0)=(0,0,...,1),$$\nand for $w\\in{\\hbox{\\mathbold\\char67}}}\\def \\sC{{\\hbox{\\smathbold\\char67}},\\,|w|<1$ we let\n$$\\theta(w)=\\arg{1-w\\over 1+w}\\in \\Big[-{\\pi\\over2},{\\pi\\over2}\\Big].$$\n\n A function depending only on $\\theta=\\sin^{-1}t^*$ can be regarded as a function on the Heisenberg sphere $\\Sigma$.\n\\medskip\\eject\n\\proclaim {Theorem 18 ([BFM], Thm. 2.2)}. Let $0<d<Q$ and $p=\\displaystyle{Q\\over d}$.  Define\n$$TF(\\zeta)=\\int_{S^n}}\\def\\wtilde{\\widetilde G(\\zeta,\\eta)F(\\eta)d\\eta,\\qquad F\\in L^p(\\Sn)$$\nwhere \n$$\\eqalign{G(\\zeta,\\eta)&=g_0\\big(\\theta(\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}})\\big)\\,d(\\zeta,\\eta)^{d-Q}+O\\big(d(\\zeta,\\eta)^{d-Q+\\epsilon}\\def\\part{\\partial_t}\\big)=\\cr&=\n2^{{d-Q\\over2}} g_0\\big(\\theta(\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}})\\big)\\,|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{d-Q\\over2}+O\\big(|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{{d-Q+\\epsilon}\\def\\part{\\partial_t\\over2}}\\big),\\quad\\zeta\\neq\\eta\\cr}$$\nfor bounded and measurable $g_0:\\big[\\!-{\\pi\\over2},{\\pi\\over2}\\big]\\to{\\hbox{\\mathbold\\char82}}$, with  \n$\\big|O\\big(|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{{d-Q+\\epsilon}\\def\\part{\\partial_t\\over2}}\\big)\\big|\\le C|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{{d-Q+\\epsilon}\\def\\part{\\partial_t\\over2}}$, some $\\epsilon>0$, and with $C$ independent of $\\zeta,\\eta$.\n\\smallskip Then, there exists $C_0>0$ such that for all $F\\in L^p(\\Sn)$\n$$\\int_{\\Sn}\\exp\\bigg[A_d\\bigg({|TF|\\over\\|F\\|_p}\\bigg)^{p'}\\bigg]d\\zeta\\le C_0\\eqno(114)$$\nwith\n$$A_d={2Q\\over\\displaystyle{\\int_\\Sigma}|g_0|^{p'} du^*}\\eqno(115)$$\nfor every $F\\in L^p(S^n)$, with ${1\\over p}+{1\\over p'}=1$. Moreover, if the function $g_0(\\theta)$  is H\\\"older continuous of order $\\sigma\\in(0,1]$ then the constant in (115) is sharp, in the sense that if it is replaced by a larger constant then there exists a sequence $F_m\\in L^p(\\Sn)$ such that the exponential integral in (114) diverges to $+\\infty$ as $m\\to\\infty$. \\par\\smallskip\n\n In [CoLu1] Cohn and Lu give a similar result in the context of the Heisenberg group, and for kernels of type $G(u)=g(u^*)|u|^{d-Q}$, i.e. without any perturbations. An $\\Hn$ version of Theorem 18 holds with virtually the same proof (in fact somewhat easier), but the two versions do not seem to be a consequence of each other.\n\\smallskip\nIn view of Theorem 1, to prove (114) it is enough to find an asymptotic estimate for  the distribution function of $G$. This is provided by the following result (which was proved in [BFM]):\\smallskip \n\\proclaim Proposition 19 {([BFM] Lemma 2.3)}. Let $G:\\Sn\\times\\Sn\\setminus\\{(\\zeta,\\zeta),\\zeta\\in\\Sn\\}\\to{\\hbox{\\mathbold\\char82}} $,  be measurable and such that \n$$G(\\zeta,\\eta)=g\\big(\\theta(\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}})\\big)\\,|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{-\\alpha}+O\\big(|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{-\\alpha+\\epsilon}\\def\\part{\\partial_t}\\big),\\quad\\zeta\\neq\\eta$$\n some bounded and measurable $g:\\big[-{\\pi\\over2},{\\pi\\over2}\\big]\\to{\\hbox{\\mathbold\\char82}}$, with  \n$\\big|O\\big(|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{-\\alpha+\\epsilon}\\def\\part{\\partial_t}\\big)\\big|\\le C|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{-\\alpha+\\epsilon}\\def\\part{\\partial_t}$, some $\\epsilon>0$, and with $C$ independent of $\\zeta,\\eta$.\nThen, for each $\\eta\\in \\Sn$ and as $s\\to+\\infty$\n$$\\big|\\{\\zeta:\\,|G(\\zeta,\\eta)|>s\\}\\big| = s^{-Q\/2\\alpha}\\,{2^{Q\/2-1}\\over Q}\\int_\\Sigma|g|^{Q\/2\\alpha}du^*+O\\big(s^{-Q\/2\\alpha-\\sigma}\\big)$$\n for a suitable $\\sigma>0$.\n\\par\nObserve that $G$ above  may not be symmetric, but has upper and lower bounds with enough symmetries, so that in effect \n$G(\\zeta,\\cdot)^*(t)$ and $G(\\cdot,\\eta)^*(t)$, have the same asymptotic expansion in $t$ (independent of $\\zeta,\\eta$).\nProposition 19  combined with Theorem 1 gives (114). We now apply Theorem 4 in order to show the sharpness statement (this part was not done in [BFM]).\n\\smallskip\n\\pf Proof  of sharpness statement of Thm 18. The proof is similar to that of Theorem~8. Let $\\zeta_m={\\hbox{\\gothic\\char78}}$ , $r_m=C_0m^{-1\/(Q-d)}<1$, so that \n$$\\big\\{\\eta:\\,|G({\\hbox{\\gothic\\char78}},\\eta)|>m\\big\\}\\subseteq E_m:=\\big\\{\\eta:|1-\\eta_{n+1}|<2r_m^2\\big\\}$$\nand let $B_m=\\big\\{\\zeta:|1-\\zn|<\\ts{1\\over4}r_m^2\\big\\}.$\n\\smallskip\n Conditions a), b), c) of Theorem 4 are met, from Proposition 19 and Remark 1 after Theorem 4, so all we need to do is show that \n$$\\int_{\\Sn\\setminus E_m} |G(\\zeta,\\eta)-G({\\hbox{\\gothic\\char78}},\\eta)|\\, |G({\\hbox{\\gothic\\char78}},\\eta)|^{p'-1}d\\eta\\le C\\,,\\qquad \\forall \\eta\\in B_m.\n$$\n\nWLOG we can assume that $G(\\zeta,\\eta)=g\\big(\\theta(\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}})\\big)\\,|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{d-Q\\over2}$, with $g=2^{d-Q\\over2}g_0$; as it will be apparent from the proof below,  an\nerror term of type $|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{{d-Q+\\epsilon}\\def\\part{\\partial_t\\over2}}$ will produce an integrable function on $\\Sn$, with uniformly bounded integral. So let us show that\n$$\\mathop\\int\\limits_{|1-\\eta_{n+1}|\\ge 2r_m^2} |G(\\zeta,\\eta)-G({\\hbox{\\gothic\\char78}},\\eta)|\\, |G({\\hbox{\\gothic\\char78}},\\eta)|^{d\\over Q-d} d\\eta\\le C\\,,\\qquad  |1-\\zn|< \\ts{1\\over4}r_m^2\n\\eqno(116)$$\n\nBy adding and subtracting the quantity \\def\\n\\cdot\\bar\\eta{{\\hbox{\\gothic\\char78}}\\cdot\\overline\\eta}\n$g\\big(\\theta(\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}})\\big)|1-\\eta_{n+1}|^{d-Q\\over2}$ we are reduced to proving the following estimates\n$$\\mathop\\int\\limits_{|1-\\eta_{n+1}|\\ge 2r_m^2} \\big|g\\big(\\theta(\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}})\\big)-g\\big(\\theta(\\n\\cdot\\bar\\eta)\\big)\\big||g(\\n\\cdot\\bar\\eta)|^{d\\over Q-d}|1-\\eta_{n+1}|^{-Q\/2}d\\eta\\le C\\eqno(117)$$\n$$\\mathop\\int\\limits_{|1-\\eta_{n+1}|\\ge2 r_m^2}\\big|g\\big(\\theta(\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}})\\big)\\big|\\big|g\\big(\\theta(\\n\\cdot\\bar\\eta)\\big)\\big|^{d\\over Q-d}\\big||1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{d-Q\\over2}-|1-\\eta_{n+1}|^{d-Q\\over2}\\big| \\,|1-\\eta_{n+1}|^{-d\/2}d\\eta\\le C\\eqno(118)$$\nvalid for all $\\zeta\\in B_m$.\n\nThe first step is to transfer these integrals to $\\Hn$ via the Cayley transform. Recall that the volume density of the Cayley transform is \n$$|J_{\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}(z,t)|={2^{2n+1}\\over \\big((1+|z|^2)^2+t^2\\big)^{n+1}}\\le {2^{2n+1}\\over (1+|u|^4)^{Q\/2}}$$\n\nIf $u=(z,t),\\,v=(z',t')$, and $\\zeta={\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}(u),\\,\\eta={\\cal C}}\\font\\mathbold=msbm9 at 10pt\\def\\Hn{{\\hbox{\\mathbold\\char72}}^n}\\def\\Sn{{S^{2n+1}}(v)$ then for $m$ large enough (using (113))\n $$2r_m^2\\le|1-\\eta_{n+1}|={2|v|^2\\over(1+2|z'|^2+|v|^4)^{1\/2}}\\le2|v|^2\\;\\;\\Longrightarrow\\;\\; |v|\\ge r_m$$\n$${r_m^2\\over4}>|1-\\zn|= {2|u|^2\\over(1+2|z|^2+|u|^4)^{1\/2}}\\ge {2|u|^2\\over1+|u|^2}\\;\\;\\Longrightarrow\\;\\; |u|<{r_m\\over2}$$\nand so if $\\eta\\notin E_m,\\,\\zeta\\in B_m$ then (using that $|v^{-1}u|$ is a distance)\n$${|1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}|^{1\/2}\\over|1-\\eta_{n+1}|^{1\/2}}={|v^{-1}u|\\over|v|}\\,{1\\over(1+2|z|^2+|u|^4)^{1\/4}}\\ge{{1-\\displaystyle{|u|\\over|v|}\\over(1+|u|^2)^{1\/2}}}\\ge{1\\over 2\\sqrt 2}.$$\nSince $|v^{-1}u|\/|v|=1+O(|u|\/|v|)$, for our range of $u$ and $v$, we obtain that the integrand in (118) in $\\Hn$ coordinates is bounded above by\n$$J_m=C\\int_{|v|\\ge r_m}\\bigg({|v|^2\\over1+|v|^2}\\bigg)^{-Q\/2} \\bigg({|u|\\over|v|}+|u|^2\\bigg){1\\over(1+|v|^4)^{Q\/2}}dv,\\,\\qquad |u|<{r_m\\over2}.\\eqno(119)$$\nThe integrand in (119) is bounded above by an integrable function on $\\{|v|\\ge 1\\}$, hence\n$$J_m\\le C+\\int_{r_m\\le|v|\\le 1}\\Big(|v|^{-Q-1}|u|+|v|^{-Q}|u|^2\\Big)dv=C\\int_{r_m}^1 \\Big(r^{-2}|u|+r^{-1}|u|^2\\Big)dr\\le C$$\nwhich proves (118).\n\nTo prove (117), if $|z|^2+it=|u|^2 e^{i\\theta}$\nand $|z'|^2+it'=|v|^2e^{i\\varphi}$, then\\defz\\cdot\\bar {z'}{z\\cdot\\overline {z'}}\n$${1-\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}\\over1+\\z\\!\\cdot\\bar\\eta}\\def\\zbn{\\z\\!\\cdot\\bar{\\hbox{\\gothic\\char78}}}={|u|^2e^{i\\theta}+|v|^2e^{-i\\varphi}-2z\\cdot\\bar {z'}\\over 1+|u|^2|v|^2e^{i(\\theta-\\varphi)}+2z\\cdot\\bar {z'}}$$\nso, since $g$ is H\\\"older continuous, (117) is implied by \n$$\\int_{r_m\\le |v|\\le 1} \\big|\\arg(1+|u|^2|v|^2e^{i(\\theta-\\varphi)}+2z\\cdot\\bar {z'})\\big|^{\\sigma}\\,|u|^{-Q}du\\le C$$\nand\n$$\\int_{r_m\\le |v|\\le 1} \\big|\\arg(|u|^2e^{i\\theta}+|v|^2e^{-i\\varphi}-2z\\cdot\\bar {z'})+\\varphi\\big|^{\\sigma}\\,|u|^{-Q}du\\le C$$\nwhenever  $|u|<{1\\over2} r_m$. Both  these estimates follow easily as above, from the simple observation that $\\arg(e^{-i\\varphi}+t \\omega)=-\\varphi+O(t)$, as $t\\to0$, if $|\\omega|\\le C$, uniformly in $\\varphi$ (recall that $-\\pi\/2\\le\\varphi\\le \\pi\/2$). This concludes the proof of (116) and the sharpness statement of Theorem  18.\n$$\\eqno\/\\!\/\\!\/$$\n\n\\vskip1.5em \n\n\n\\centerline{\\bf References}\\bigskip\n\\item{[Ad1]} Adams D.R. {\\sl\nA sharp inequality of J. Moser for higher order derivatives},\nAnn. of Math. {\\bf128} (1988), no. 2, 385--398. \n\\smallskip\n\\item{[Ad2]} Adams D.R.\n{\\sl Traces of potentials arising from translation invariant operators}, \nAnn. Scuola Norm. Sup. Pisa (3) {\\bf 25} (1971) 203-217. \n\\smallskip\n\\item{[Ad3]} Adams D.R., {\\sl A trace inequality for generalized potentials},\nStudia Math. {\\bf 48} (1973), 99--105. \n\\smallskip\n\\item{[Au1]} Aubin T., {\\sl Probl\\`emes isop\\'erim\\'etriques at espaces de Sobolev}, J. 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Z. {\\bf 250} (2005), 363-386.\\smallskip \n\\item{[MMcO]} Maz'ya V., McOwen R.C., {\\sl On the fundamental solution of an elliptic equation in nondivergence form}, arXiv:0806.4108v1.\\smallskip\n\\item{[Mi]} Miranda C., {\\sl Partial Differential Equations of Elliptic Type}, Springer-Verlag, New York (1970).\\smallskip\n\\item{[Mil]} Milman E., {\\sl Generalized Intersection Bodies}, J. Funct. Anal. {\\bf 240} 2006, 530-567.\\smallskip\n\\item{[Mos1]} Moser J. {\\sl A sharp form of an inequality by N. Trudinger}, Indiana Univ. Math. J. {\\bf20} (1970\/71), 1077-1092.\\smallskip\n\\item{[Mos2]}  Moser J., {\\sl On a nonlinear problem in differential geometry}, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), 273-280. Academic Press, New York, 1973. \n\\smallskip\n\\item{[ON]}O'Neil R., {\\sl Convolution operators in $L(p,q)$ spaces}, Duke Math. J. {\\bf 30} (1963), 129-142. \\smallskip\n\\item{[R]} Ruf B., {\\sl A sharp Trudinger--Moser type inequality for unbounded domains in ${\\hbox{\\mathbold\\char82}}^2$}, J. Funct. Anal. {\\bf 219} (2005), 340-367.\\smallskip\n\\item{[Tr]} Trudinger N.S., {\\sl \nOn imbeddings into Orlicz spaces and some applications}, \nJ. Math. Mech. {\\bf17} (1967) 473-483. \n\\smallskip\n\\item{[Z]} Zygmund A., {\\sl Trigonometric Series}, Vol. 1, Cambridge Univ. Press (1959).\n\n\n\\bigskip\n\n\\noindent Luigi Fontana \\hskip19em Carlo Morpurgo\n\n\\noindent Dipartimento di Matematica ed Applicazioni \\hskip5.5em Department of Mathematics \n \n\\noindent Universit\\'a di Milano-Bicocca\\hskip 13em University of Missouri, Columbia\n\n\\noindent Via Cozzi, 53 \\hskip 19.3em Columbia, Missouri 65211\n\n\\noindent 20125 Milano - Italy\\hskip 16.6em USA \n\\smallskip\\noindent luigi.fontana@unimib.it\\hskip 15.3em morpurgoc@missouri.edu\n\n\\end\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA semigroup $O$ or an associative algebra $O$ is called {\\em nilpotent} if \nthere exists an integer $c$ so that every product of $c+1$ elements equals \nzero. The least integer $c$ with this property is the {\\em class} $cl(O)$ \nof $O$; equivalently, the class of $O$ is the length of series of powers\n\\[ O > O^2 > \\ldots > O^c > O^{c+1} = \\{0\\}. \\]\nThe {\\em coclass} of a finite nilpotent semigroup $O$ with $n$ non-zero \nelements or a finite dimensional nilpotent algebra $O$ of dimension $n$ \nis defined via $cc(O) = n-cl(O)$.\n\nFor a semigroup $S$ and a field $K$ we denote with $K[S]$ the semigroup \nalgebra defined by $K$ and $S$. This is an associative algebra of dimension\n$|S|$. If $S$ has a zero element $z$, then the subspace $U$ of $K[S]$ \ngenerated by $z$ is an ideal in $K[S]$. We call $K[S]\/U$ the {\\em \ncontracted semigroup algebra} defined by $K$ and $S$ and denote it by \n$KS$. If $S$ is a finite nilpotent semigroup, then $KS$ is a nilpotent \nalgebra of the same class and coclass as $S$.\n\nOur first aim in this note is to suggest a general approach towards a\nclassification up to isomorphism of nilpotent semigroups of a fixed \ncoclass. For this purpose we choose an arbitrary field $K$ and we define\na directed labelled graph $\\mathcal G_{r, K}$ as follows: the vertices of $\\mathcal G_{r, K}$ \ncorrespond one-to-one to the isomorphism types of algebras $KS$ for the \nnilpotent semigroups $S$ of coclass $r$; two vertices $A$ and $B$ are \nadjoined by a directed edge $A \\rightarrow B$ if $B\/B^c \\cong A$, where $c$ is \nthe class of $B$; each vertex $A$ in $\\mathcal G_{r,K}$ is labelled by the \nnumber of isomorphism types of semigroups $S$ of coclass $r$ \nwith $A \\cong KS$. Illustrations of parts of such graphs can be found \nas Figure~\\ref{figcc1} on page~\\pageref{cc1} and as Figure~\\ref{figcc2d2}\non~\\pageref{cc2}.\n\nWe have investigated various of the graphs $\\mathcal G_{r,K}$ and we observed that\nall these graphs share the same general features. We formulate a sequence of \nconjectures and theorems describing these features. If our conjectures become \ntheorems, then this would provide the ground for a new approach towards the \nclassification and investigation of nilpotent semigroups by coclass. In \nparticular, it would show how the classification of the infinitely many \nnilpotent semigroups of a fixed coclass reduces to a finite calculation.\n\nAs a second aim in this note we exhibit some graphs $\\mathcal G_{r,K}$ explicitly \nto illustrate our conjectures. We have determined the graphs $\\mathcal G_{0,K}$ and \n$\\mathcal G_{1,K}$ for all fields $K$ using the classification of the nilpotent \nsemigroups of small coclass in \\cite{Dis10, Dis11}; see Sections \n\\ref{cc0} and \\ref{cc1}. Further, we investigated the graphs $\\mathcal G_{2,K}$ and \n$\\mathcal G_{3,K}$ for some finite fields $K$ using computational methods based on \n\\cite{Eic07} to solve the isomorphism problem for nilpotent associative \nalgebras over finite fields; see Section \\ref{cc2}.\n\nSimilar to the graphs $\\mathcal G_{r,K}$ one can also define a directed graph $\\mathcal G_r$ \nwhose vertices correspond one-to-one to the isomorphism types of semigroups of \ncoclass $r$. While the graphs $\\mathcal G_r$ are also of interest, they do not\nexhibit the same general features as $\\mathcal G_{r,K}$. We compare $\\mathcal G_r$ and\n$\\mathcal G_{r,K}$ briefly in Section \\ref{final}.\n\nThe idea of using the coclass for the classification of nilpotent algebraic\nobjects has first been introduced by Leedham-Green \\& Newman \\cite{LNe80}\nfor nilpotent groups. We also refer to the book by Leedham-Green \\& McKay\n\\cite{LGM02} for background and many details on the results in the group \ncase. Various details of the approach taken here are similar to the \nconcepts in group theory. In particular, the idea of searching for periodic \npatterns in coclass graphs as used below also arises in group theory; we \nrefer to \\cite{DuS01, ELG08} for details. Note though that a nilpotent\nsemigroup is not a group and hence the coclass theories for groups and\nsemigroups are independent.\n\n\\section{Coclass conjectures for semigroups} \n\\label{conj}\n\nIn this section we investigate general features of the graph $\\mathcal G_{r,K}$\nfor $r \\in \\mathbb N_0$ and arbitrary field $K$.\n\nBy construction, every connected component of $\\mathcal G_{r,K}$ is a rooted tree.\nUsing basic results on nilpotent semigroups (see~\\cite[Lemma 2.1]{Dis11})\none readily shows that $2r$ is an upper bound for the dimension of a root\n(that is, the dimension of the corresponding algebra) in $\\mathcal G_{r,K}$. Thus\n$\\mathcal G_{r,K}$ consists of finitely many rooted trees. We call an infinite path\nin a rooted tree {\\em maximal} if it starts at the root of the tree.\n\n\\begin{conjecture}\n\\label{conjA}\nLet $r \\in \\mathbb N_0$ and $K$ an arbitrary field. \nThen the graph $\\mathcal G_{r,K}$ has only finitely many maximal infinite paths. \nThe number of such paths depends on $r$ but not on $K$.\n\\end{conjecture}\n\nFor an algebra $A$ in $\\mathcal G_{r,K}$ we denote by $\\mathcal T(A)$ the subgraph of\n$\\mathcal G_{r,K}$ consisting of all paths that start at $A$. This is a rooted\ntree with root $A$. We say that $\\mathcal T(A)$ is a {\\em coclass tree} \nif it contains a unique maximal infinite path. A coclass tree\n$\\mathcal T(A)$ is {\\em maximal} if either $A$ is a root in $\\mathcal G_{r,K}$ or\nthe parent of $A$ lies on more than one maximal infinite paths.\n\n\\begin{remark}\nConjecture \\ref{conjA} is equivalent to saying that $\\mathcal G_{r,K}$ consists \nof finitely many maximal coclass trees and finitely many other vertices.\n\\end{remark}\n\nWe consider the maximal coclass trees in $\\mathcal G_{r,K}$ in more detail. For a\nlabelled tree $\\mathcal T$ we denote with $\\overline{\\mathcal T}$ the tree without labels.\n\n\\begin{conjecture}\n\\label{conjB}\nLet $r \\in \\mathbb N_0$ and $K$ an arbitrary field. \nLet $\\mathcal T$ be a maximal coclass tree in $\\mathcal G_{r,K}$ with maximal infinite path\n$A_1 \\rightarrow A_2 \\rightarrow \\ldots$ Then $\\mathcal T$ is {\\em weakly virtually periodic}; that \nis, there exist positive integers $l$ and $k$ so that $\\overline{\\mathcal T}(A_l) \\cong \n\\overline{\\mathcal T}(A_{l+k})$ holds.\n\\end{conjecture}\n\nThe integers $l$ and $k$ with the property of Conjecture \\ref{conjB} are\ncalled {\\em weak defect} and {\\em weak period} of $\\overline{\\mathcal T}$. Note \nthat they are not unique. Every integer larger than $l$ and every multiple \nof $k$ are weak defects and weak periods as well, respectively.\n\nConsider a maximal coclass tree $\\mathcal T$ of $\\mathcal G_{r,K}$ with maximal\ninfinite path $A_1 \\rightarrow A_2 \\rightarrow \\ldots$ Suppose that for some $l$ and\n$k$ there exists a graph isomorphism $\\mu : \\overline{\\mathcal T}(A_l) \\rightarrow\n\\overline{\\mathcal T}(A_{l+k})$. Then $\\mu$ defines a partition of the vertices of\n$\\mathcal T(A_l)$ into finitely many infinite families: for each vertex $B$\ncontained in $\\mathcal T(A_l) \\setminus \\mathcal T(A_{l+k})$ define the infinite\nfamily $(B, \\mu(B), \\mu^2(B), \\ldots)$. Hence Conjecture \\ref{conjB}\nasserts that the unlabelled tree $\\overline{\\mathcal T}$ can be constructed from a\nfinite subgraph, provided that a weak defect and a weak period are\nknown. This implies that $\\mathcal T$ has finite width. Conjecture \\ref{conjA} \nadds that these features of maximal coclass trees extend to all of \n$\\mathcal G_{r,K}$.\nWe next exhibit an extension of Conjecture \\ref{conjB} incorporating\nlabels.\n\n\\begin{conjecture}\n\\label{conjC}\nLet $r \\in \\mathbb N_0$ and $K$ an arbitrary field. \nLet $\\mathcal T$ be a maximal coclass tree in $\\mathcal G_{r,K}$ with maximal infinite path\n$A_1 \\rightarrow A_2 \\rightarrow \\ldots$ Then $\\mathcal T$ is {\\em strongly virtually periodic}; \nthat is, there exist positive integers $l$ and $k$, a graph isomorphism \n$\\mu : \\overline{\\mathcal T}(A_l) \\rightarrow \\overline{\\mathcal T}(A_{l+k})$ and for every vertex $B$ \nin $\\mathcal T(A_l) \\setminus \\mathcal T(A_{l+k})$ a rational polynomial $f_B$ so that \nthe label of $\\mu^i(B)$ equals $f_B(i)$.\n\\end{conjecture}\n\nThe integers $l$ and $k$ with the property of Conjecture \\ref{conjC} are\ncalled {\\em strong defect} and {\\em strong period} of $\\mathcal T$. As in the\nweak case, they are not unique. Further, every strong defect and strong\nperiod are also a weak defect and weak period, but the converse does not\nhold in general; compare Section \\ref{cc1} for an example.\n\nConjectures \\ref{conjA} and \\ref{conjC} suggest the following new approach \ntowards a classification up to isomorphism of all nilpotent semigroups of \nfixed coclass $r \\in \\mathbb N_0$. \n\n\\begin{items}\n\\item[(1)]\nChoose an arbitrary field $K$ and classify the maximal infinite paths in \n$\\mathcal G_{r,K}$.\n\\item[(2)]\nFor each maximal infinite path consider its corresponding coclass tree\n$\\mathcal T$ and find a strong defect $l$, a strong period $k$ and an upper bound \n$d$ to the degree of the polynomials of the associated families. \n\\item[(3)]\nFor each maximal coclass tree $\\mathcal T$ with strong defect $l$, strong period $k$ \nand bound $d$:\n\\begin{items}\n\\item[(a)]\nDetermine the unlabelled tree $\\overline{\\mathcal T}$ up to depth $l+(d+1)k$. \n\\item[(b)]\nFor each vertex $B$ in the determined part of $\\overline{\\mathcal T}$ compute its\nlabel.\n\\end{items}\n\\item[(4)]\nDetermine the finite parts of $\\mathcal G_{r,K}$ outside the maximal coclass trees.\n\\end{items}\n\nStep (1) is discussed further in Section \\ref{infdim} below. For Step (2)\nit would be the hope that a constructive proof of the conjectures posed\nhere might also yield values for strong defect, strong period and bounds \nfor the degrees of the arising polynomials.\n\nSteps (3a) and (3b) may be facilitated by two algorithms. The first \ndetermines up to isomorphism all contracted semigroup algebras $B$ of \nclass $c+1$ with $B\/B^{c+1} \\cong A$ for any given contracted semigroup \nalgebra $A$ of class $c$. The second algorithm takes a nilpotent \nassociative algebra $A$ of finite dimension and computes up to isomorphism \nall semigroups $S$ with $K S \\cong A$. Both algorithms reduce to a finite\ncomputation if the underlying field $K$ is finite. A practical realisation\nfor the first algorithm in the finite field case may be obtained as \nvariation of the method in \\cite{Eic07}.\n\nOnce the Steps (1) - (4) have been performed, this would allow to construct \nthe full graph $\\mathcal G_{r,K}$ using the graph isomorphism of Conjecture\n\\ref{conjC}. The polynomials $f_B$ can be interpolated from the given \ninformation, as there are $d+1$ values $f_B(i)$ available.\n\n\\section{The infinite paths in $\\mathcal G_{r,K}$}\n\\label{infdim}\n\nIn this section we investigate in more detail the infinite paths in \n$\\mathcal G_{r,K}$ for arbitrary $r \\in \\mathbb N_0$ and arbitrary field $K$. We first \nprovide some background for our constructions.\n\n\\subsection{Coclass for infinite objects}\n\nLet $O$ be a finitely generated infinite semigroup or a finitely generated \ninfinite dimensional associative algebra. Then every quotient $O\/O^i$ is \nfinitely generated of class at most $i$ and hence is finite (in the semigroup\ncase) or finite dimensional (in the algebra case). Thus $O\/O^i$ has finite\ncoclass $cc(O\/O^i)$. We say that $O$ is {\\em residually nilpotent} if \n$\\cap_{i \\in \\mathbb N} O^i = 0$ holds. If $O$ is finitely generated and \nresidually nilpotent, then we define its {\\em coclass} $cc(O)$ as\n\\[ cc(O) = \\lim_{i \\rightarrow \\infty} cc(O\/O^i).\\]\n\nThe coclass of $O$ can be finite or infinite. It is finite if and only if\nthere exists $i \\in \\mathbb N$ so that $|O^{j+1} \\setminus O^j| = 1$ (in the\nsemigroup case) or $\\dim(O^j\/O^{j+1}) = 1$ (in the algebra case) for all \n$j \\geq i$. If we say that $O$ has `finite coclass', then this implies\nthat $O$ is finitely generated and residually nilpotent.\n\n\\subsection{Inverse limits of algebras and semigroups}\n\nConsider a maximal infinite path $A_1 \\rightarrow A_2 \\rightarrow \\ldots$ in $\\mathcal G_{r,K}$ and\nlet $\\hat{A} = \\prod_{i \\in \\mathbb N} A_i$ be the Cartesian product of the algebras\non the path. If $A_1$ has class $c$, then $A_j$ has class $j+c-1$ and thus\n$A_{j+1}\/A_{j+1}^{j+c} \\cong A_j$ for every $j \\in \\mathbb N$. For every $j \\in \\mathbb N$ \nwe choose an epimorphism $\\nu_j : A_{j+1} \\rightarrow A_j$ with kernel \n$A_{j+1}^{j+c}$. We define the {\\em inverse limit} of the algebras on the path as\n\\[ A = \\left\\{ (a_1, a_2, \\ldots) \\in \\hat{A} \\mid \\nu_j(a_{j+1}) = a_j \n     \\mbox{ for every } j \\in \\mathbb N \\right\\}.\\]\n\nThe inverse limit $A$ is an infinite dimensional associative $K$-algebra\nwhich satisfies $A \/ A^{j+c} \\cong A_j$ for every $j \\in \\mathbb N$. Thus $A\/A^2$\nis finite dimensional and hence $A$ is finitely generated. It is also \nresidually finite and has coclass $r$. Further, each algebra on the maximal \ninfinite path can be obtained as quotient of $A$ and thus $A$ fully \ndescribes the considered maximal infinite path. We summarize this as follows.\n\n\\begin{theorem}\n\\label{infpath}\nLet $r \\in \\mathbb N_0$ and $K$ an arbitrary field.\nFor every maximal infinite path in $\\mathcal G_{r,K}$ there exists an infinite\ndimensional associative $K$-algebra of coclass $r$ which describes \nthe path. \n\\end{theorem}\n\nIsomorphic algebras of the type considered in Theorem \\ref{infpath} \ndescribe the same infinite path. Hence an approach to the classification \nof the maximal infinite paths in $\\mathcal G_{r,K}$ is the determination up to \nisomorphism of the infinite dimensional associative $K$-algebras $A$ of \ncoclass $r$ whose quotients $A\/A^j$ are contracted semigroup algebras\nfor every $j \\in \\mathbb N$. Conjecture \\ref{conjA} is equivalent to saying that\nthere are only finitely many of these objects up to isomorphism. The\nfollowing theorem describes these algebras in more detail.\n\n\\begin{theorem}\nLet $r \\in \\mathbb N_0$ and $K$ an arbitrary field.\nEach infinite dimensional associative $K$-algebra of coclass $r$ which\ndescribes an infinite path in $\\mathcal G_{r,K}$ is isomorphic to a contracted \nsemigroup algebra $KS$ for an infinite semigroup $S$ of coclass $r$.\n\\end{theorem}\n\n\\begin{proof}\nLet $A$ be an infinite dimensional associative $K$-algebra of coclass $r$\nwhich describes an infinite path in $\\mathcal G_{r,K}$. Then there exists an $i \\in \n\\mathbb N$ so that $A\/A^j$ is a contracted semigroup algebra of coclass $r$ for \nevery $j \\geq i$. Each of the quotients $A\/A^j$ may be the contracted \nsemigroup algebra for several non-isomorphic semigroups. Our aim is to show \nthat for every $j \\geq i$ there exists a semigroup $S_j$ so that $A\/A^j \n\\cong K S_j$ and $S_j \\cong S\/S^j$ for an infinite semigroup $S$ of \ncoclass $r$.\n\nWe define a graph $\\L$ whose vertices correspond one-to-one to the isomorphism\ntypes of semigroups whose contracted semigroup algebra is isomorphic to a\nquotient $A\/A^j$ for some $j \\geq i$. We connect two semigroups in $\\L$\nby a directed edge $U \\rightarrow T$ if $T\/T^c \\cong U$, where $c$ is the class of\n$T$. If a semigroup $T$ satisfies $KT \\cong A\/A^j$ for some $j > i$, then \n$T$ has class $j-1$ and $U \\cong T\/T^{j-1}$ satisfies $KU \\cong A\/A^{j-1}$.\nHence each connected component of $\\L$ is a tree with a root of class $i-1$\nand coclass $r$. There is at least one infinite connected component $\\mathcal M$ of\n$\\L$. By K\\\"onig's Lemma, the tree $\\mathcal M$ contains an infinite path, say\n$M_i \\rightarrow M_{i+1} \\rightarrow \\ldots$ Let $S$ be the inverse limit of the semigroups\non this infinite path. Then $S$ is an infinite semigroup with $S\/S^j\n\\cong M_j$ and $KM_j \\cong A\/A^j$ for every $j \\geq i$. In particular, the\nsemigroup $S$ has finite coclass $r$. \n\nIt remains to show that $S$ satisfies $KS \\cong A$. This follows from the\nconstruction of $S$, as the following diagram is commutative, where upwards\narrows denote embeddings of semigroups in their contracted semigroup \nalgebras:\n\\[ \\begin{array}{ccccccc}\n   A\/A^i & \\rightarrow & A\/A^{i+1} & \\rightarrow & A\/A^{i+2} & \\rightarrow & \\ldots \\\\\n   \\uparrow & & \\uparrow & & \\uparrow && \\\\\n   S\/S^i & \\rightarrow & S\/S^{i+1} & \\rightarrow & S\/S^{i+2} & \\rightarrow & \\ldots \\\\\n   \\end{array}. \\]\nThis completes the proof.\n\\end{proof}\n\n\\subsection{Examples}\n\nConsider the polynomial algebra in one indeterminate and let $I_K$ denote\nits ideal consisting of all polynomials with zero constant term. Then $I_K$\nis an explicit construction for the free non-unital associative algebra\non one generator over the field $K$. It is isomorphic to the contracted\nsemigroup algebra $KS$ with $S \\cong (\\mathbb N_0, +)$ and it has coclass $0$. \nHence it describes an infinite path in $\\mathcal G_{0,K}$. In Section \\ref{cc0} \nwe observe that it describes the unique maximal infinite path in $\\mathcal G_{0,K}$.\n\nExamples of infinite dimensional contracted semigroup algebras of higher \ncoclass can be obtained inductively using the following process. Let $S$\nbe an infinite semigroup of coclass $r-1$. An {\\em annihilator extension} \nof $S$ is an infinite semigroup $T$ so that $T$ contains a non-zero \nelement $t \\in \\operatorname{Ann}(T)$ with $S \\cong T \/ \\langle t \\rangle$. \n\n\\begin{lemma}\nLet $r \\in \\mathbb N$ and let $S$ be an infinite semigroup of coclass $r-1$. \nEach annihilator extension $T$ of $S$ is an infinite semigroup of coclass \n$r$.\n\\end{lemma}\n\n\\begin{proof}\nConsider the sequence $T \\geq T^2 \\geq T^3 \\geq \\ldots$ and define $c \\in \\mathbb N$\nvia $t \\in T^c \\setminus T^{c+1}$. Let $\\nu : T \\rightarrow S$ be an epimorphism with \nkernel $\\langle t \\rangle$. As $\\nu(T^i) = S^i$, we obtain that $T\/T^i \\cong \nS\/S^i$ for $1 \\leq i \\leq c$ and for $i \\geq c+1$ we obtain that $|T\/T^i| = \n|S\/S^i|+1$. Thus $T$ is finitely generated and residually nilpotent and it \nsatisfies $cc(T\/T^i) = cc(S\/S^i)+1$ for $i \\geq c+1$. Thus $T$ is an infinite\nsemigroup of coclass $cc(T) = cc(S)+1$.\n\\end{proof}\n\nIf $A$ is an infinite dimensional contracted semigroup algebra of coclass \n$r-1$, then $A = KS$ for an infinite semigroup $S$ of coclass $r-1$. Thus \nevery annihilator extension $T$ of $S$ gives rise to an infinite dimensional\ncontracted semigroup algebra of coclass $r$. \n\nWe exhibit an explicit example for this process. For this purpose let $L_K$ \ndenote the 1-dimensional nilpotent algebra of class $1$. Then $L_K$ is \nisomorphic to the contracted semigroup algebra $KZ_2$, where $Z_n$ is the\nzero semigroup with $n$ elements. For every $r\\in\\mathbb N$ the algebra \n\\begin{equation}\n\\label{eq_MKr}\nM_{K,r} = I_K \\oplus \\bigoplus_{i=1}^r L_K\n\\end{equation}\nis an infinite dimensional contracted semigroup algebra of coclass\n$r$. As underlying semigroup one can choose the zero union of $(\\mathbb N,+)$\nand $Z_{r+1}$, that is the semigroup on $\\mathbb N \\cup Z_{r+1}$ in which\nmixed products equal $0\\in Z_{r+1}$. For $r>0$ this is an annihilator\nextension of the zero union of  $(\\mathbb N,+)$ and $Z_{r}$ corresponding to\n$M_{K,r-1}$.\n\nWe close this section by posing the following question.\n\n\\begin{question}\n\\label{conjE}\nDoes every infinite dimensional algebra which describes an infinite\npath in $\\mathcal G_{r,K}$ arise as contracted semigroup algebra for a semigroup\nwhich is an annihilator extension?\n\\end{question}\n\n\n\\section{The minimal generator number}\n\\label{sec_mingen}\n\nA nilpotent semigroup $S$ has a unique minimal generating set $S \\setminus\nS^2$. Its cardinality corresponds to the dimension of the quotient $KS \/ \n(KS)^2$ and thus to the minimal generator number of the algebra $KS$. Hence\n$KS \\cong KT$ implies that the nilpotent semigroups $S$ and $T$ have the\nsame minimal generator number. Further, if two algebras in $\\mathcal G_{r,K}$ are \nconnected, then they have the same minimal generator number. This allows \nto define the subgraph $\\mathcal G_{r,K,d}$ of $\\mathcal G_{r,K}$ corresponding to the \nnilpotent semigroups of coclass $r$ with minimal generator number $d$.\n\nA nilpotent semigroup of coclass $r$ has at most $r+1$ generators. \nThus $\\mathcal G_{r,K,d}$ is empty for $d \\geq r+2$ (and also for $d=1$ if $r > 0$). \nThe extremal case $\\mathcal G_{r,K,r+1}$ can be described in more detail as the \nfollowing theorem shows. Recall that $Z_n$ is the zero semigroup with $n$ \nelements and $M_{K,r}$ is defined in~\\eqref{eq_MKr}.\n\n\\begin{theorem}\n\\label{mingen}\nLet $r \\in \\mathbb N_0$ and $K$ an arbitrary field. Then \n$\\mathcal G_{r,K,r+1}$ consists of a unique maximal coclass tree with corresponding \ninfinite dimensional algebra $M_{K,r}$. The root of the maximal coclass tree\nis $K Z_{r+2}$ if $r > 0$ and $K Z_{1}$ if $r = 0$.\n\\end{theorem}\n\n\\begin{proof}\nThe semigroup $Z_{r+2}$ has $r+2$ elements, minimal generator number $r+1$, \nclass $1$ and thus coclass $r$. If $r > 0$, then $Z_{r+2}$ is the unique \nsemigroup of coclass $r$ and order at most $r+2$ and hence $KZ_{r+2}$ is a \nroot in $\\mathcal G_{r,K,r+1}$. If $r = 0$, then $Z_1$ is a root of $\\mathcal G_{0,K,1}$.\n\nIn the following we assume that $r > 0$. The case $r=0$ is similar and we\nleave it to the reader. Let $S$ be an arbitrary semigroup of class $c$ such \nthat $KS$ is in $\\mathcal G_{r,K,r+1}$. We show by induction on $|S|$ that there \nexists a path from $KZ_{r+2}$ to $KS$. As $Z_{r+2}$ is the only semigroup \nof coclass $r$, order at most $r+2$ and minimal generator number $r+1$, we \nmay assume that $|S| > r+2$. From $|S \\setminus S^2|=r+1$ follows $|S^2|=c$ \nand hence $|S^c|=2$. Thus $S\/S^c$ is a semigroup of coclass $r$ with minimal \ngenerator number $r+1$ and with $|S| - 1$ elements. This yields that there \nis an edge from $KS\/(KS)^c \\cong K (S\/S^c)$ to $KS$. By induction, there \nexists a path from $KZ_{r+2}$ to $K (S\/S^c)$ and hence to $KS$. This proves \nthat  $\\mathcal G_{r,K,r+1}$ is connected.\n\nThe infinite dimensional algebra $M_{K,r}$ has coclass $r$ and minimal\ngenerator number $r+1$ and it is a contracted semigroup algebra. It defines \na maximal infinite path in $\\mathcal G_{r,K,r+1}$. It remains to show that this \nmaximal infinite path is unique. Let $A$ be an arbitrary infinite dimensional \nassocative algebra of coclass $r$ with $r+1$ generators. Then $\\dim(A\/A^2) \n= r+1$ and $\\dim(A^i\/A^{i+1}) = 1$ for every $i \\geq 2$. Let $v,w,x\\in A$ \nsuch that $vwxA^4$ generates $A^3\/A^4$. Then both $vwA^3$ and $wxA^3$ \ngenerate $A^2\/A^3$ and hence $vw = kwx$ for some $k\\in K$. This yields \n$vwx=kwxx$ and hence $x^2A^3$ is a generator of $A^2\/A^3$. By induction, \nit follows that $x^iA^{i+1}$ is a generator of $A^i\/A^{i+1}$ for every \n$i\\geq 2$. Now choose elements $x_1, \\ldots, x_r \\in A$ that together with \n$x$ correspond to a basis of $A\/A^2$. Then these elements generate $A$. \nA basis of $A^2$ has the form $\\{ x^j \\mid j \\geq 2\\}$. Thus for $i \\in \n\\{1, \\ldots, r\\}$ we find that \n\\[ x x_i = \\sum_{j=2}^\\infty k_{ij} x^j \\in A^2. \\]\nWe replace $x_i$ by \n\\[ y_i = x_i - \\sum_{j=2}^\\infty k_{ij} x^{j-1}\\]\nand thus obtain a new minimal generating set $x, y_1, \\ldots, y_r$ of $A$ which\nsatisfies $x y_i = 0$ by construction. For $i,j \\in \\{1,\\ldots, r\\}$\nand consider the product $y_i y_j$. Then $y_i y_j = \\sum_{h=2}^\\infty k_h x^h \n\\in A^2$. As $x y_i = 0$, it follows that $x y_i y_j = 0$ and thus \n$\\sum_{h=2}^\\infty k_h x^{h+1} = 0$. This implies that all coefficients \n$k_h$ equal $0$ and hence $y_i y_j = 0$ for every $i, j \\in \\{1, \\ldots, r\\}$. \nThis yields that $A \\cong M_{K,r}$.\n\\end{proof}\n\n\\section{The graph $\\mathcal G_{0,K}$}\n\\label{cc0}\n\nThe semigroups of coclass $0$ are well-known; for every order $n\\in\\mathbb N$ there\nexists exactly one such semigroup with presentation $\\langle u\\mid \nu^n=u^{n+1}\\rangle$. Together with the result from\nTheorem~\\ref{mingen} this leads to the following theorem.\n\n\\begin{theorem}\nLet $K$ be an arbitrary field. The graph $\\mathcal G_{0,K}$ consists of a\nunique maximal coclass tree with root $K \\!Z_1$. This tree is strongly\nvirtually periodic with strong defect $1$, strong period $1$, and the\nsingle associated polynomial $f_{K\\!Z_1}(x) = 1$.\n\\end{theorem}\n\n\\section{The graph $\\mathcal G_{1,K}$}\n\\label{cc1}\n\nWe determine the graph $\\mathcal G_{1,K}$ for arbitrary fields $K$ using the \nclassification \\cite{Dis10, Dis11} of nilpotent semigroups of coclass $1$.\nAs preliminary step, note that a nilpotent semigroup of coclass $1$ has \nat least $3$ elements. Up to isomorphism there exist exactly one\nsemigroup of coclass  $1$ with $3$ elements, namely $Z_3$, and nine\nsemigroups with $4$ elements.\n\\begin{theorem}\nLet $K$ be an arbitrary field. \n\\begin{items}\n\\item[\\rm (1)]\nThe graph $\\mathcal G_{1,K}$ consists of a unique maximal coclass tree $\\mathcal T$\nwith root $K \\!Z_{3}$ and corresponding infinite dimensional algebra\n$M_{K,1}$ (defined in \\eqref{eq_MKr}).\n\\item[\\rm (2)]\nThe tree $\\mathcal T$ is strongly virtually periodic with strong defect $2$ and \nstrong period $2$. Let $A_1 \\rightarrow A_2 \\rightarrow \\ldots$ denote the maximal infinite \npath of $\\mathcal T$. For each algebra $B\\in \\mathcal T(A_2)\\setminus\\mathcal T(A_4)$\nthe polynomial corresponding to $B$ has degree at most $1$.\n\\begin{items}\n\\item[\\rm (a)]\nIf $\\sqrt{-1} \\in K$, then $\\mathcal T(A_2)\\setminus\\mathcal T(A_4)$ consist of $A_2$,\n$4$ algebras with $A_2$ as parent, and $3$ algebras with $A_3$ as\nparent; see the right box of Figure~\\ref{figcc1}. \n\\item[\\rm (b)]\nIf $\\sqrt{-1} \\in K$, then $\\mathcal T(A_2)\\setminus\\mathcal T(A_4)$ consist of $A_2$,\n$4$ algebras with $A_2$ as parent, and $4$ algebras with $A_3$ as\nparent; see the left box of Figure~\\ref{figcc1}. \n\\end{items}\n\\end{items}\n\\end{theorem}\n\n\\begin{figure}[thb]\n\\begin{center}\n\\includegraphics[scale=0.5]{cc1.png}\n\\end{center}\n\\vspace{-1cm}\n\\caption{Description of $\\mathcal T(A_2)$ in $\\mathcal G_{1,K}$ with root of \ndimension $3$. Vertices with box correspond to non-commutative algebras.\nThe polynomials of degree $1$ are $2x+2$ and $2x+3$ for the two families \non the infinite path and $x+2$ for the other two families.}\n\\label{figcc1}\n\\end{figure}\n\n\\begin{proof}\nThe first part of the statement is true by Theorem \\ref{mingen}. To prove \nthe second part we use the classification from \\cite{Dis11}: there are the \nfollowing $n+2+ \\lfloor n\/2 \\rfloor$  isomorphism types of semigroups of \norder $n$ and coclass $1$ for $n \\geq 5$:\n\\begin{items}\n\\item[$\\bullet$] \n$H_k = \\langle u,v \\mid \nu^{n-1}=u^n, uv=u^k, vu=u^k,v^2=u^{2k-2} \\rangle, 2\\leq k\\leq n-1$;\n\\item[$\\bullet$]\n$J_k=\\langle u,v \\mid \nu^{n-1}=u^n, uv=u^k, vu=u^k,v^2=u^{n-2} \\rangle, n\/2 < k\\leq n-1$;\n\\item[$\\bullet$]\n$X = \\langle u,v \\mid u^{n-1}=u^n, uv=u^{n\/2}, vu=u^{n\/2},v^2=u^{n-1}\n  \\rangle, \\mbox{ if }n \\equiv 0 \\mod 2$;\n\\item[$\\bullet$]\n$N_1=\\langle u,v \\mid u^{n-1}=u^n, uv=u^{n-1}, vu=u^{n-2},v^2=u^{n-2} \\rangle$;\n\\item[$\\bullet$]\n$N_2=\\langle u,v \\mid u^{n-1}=u^n, uv=u^{n-2}, vu=u^{n-1},v^2=u^{n-2} \\rangle$;\n\\item[$\\bullet$]\n$N_3=\\langle u,v \\mid u^{n-1}=u^n, uv=u^{n-1}, vu=u^{n-2},v^2=u^{n-1} \\rangle$;\n\\item[$\\bullet$]\n$N_4=\\langle u,v \\mid u^{n-1}=u^n, uv=u^{n-2}, vu=u^{n-1},v^2=u^{n-1} \\rangle$.\n\\end{items}\n\nWe now show which of these semigroups give rise to isomorphic algebras.\n\\medskip \n\nShow that $K H_2 \\cong K H_k$ for $3 \\leq k \\leq n-1$ holds.\\\\\nDefine $\\mu : K H_2 \\rightarrow K H_k$ via $\\mu(u) = u+u^{k-1}$ and $\\mu(v) = u+v$.\nAs $(u+u^{k-1})^m = \\sum_{i=0}^m {m \\choose i} u^{m-i} (u^{k-1})^i = \n\\sum_{i=0}^m {m \\choose i} u^{m+(k-2)i}$ for $1\\leq m\\leq n-1$, it follows\nthat the elements $u+u^{k-1}$ and $u+v$ generate $K H_k$. The images of \n$u$ and $v$ under $\\mu$ satisfy the relations of $H_2$ and hence $\\mu$ \ninduces an epimorphism. As $K H_k$ and $K H_2$ have the same dimension, \nit follows that $\\mu$ is an isomorphism.\n\\medskip\n\nShow that $K J_{n-1} \\cong K J_k$ for $n\/2 < k < n-1$ and \n$K X \\cong K J_{n-1}$ if $n$ is even and $\\sqrt{-1} \\in K$ hold.\\\\\nFor the first part, define $\\mu : K J_{n-1} \\rightarrow K J_k$ via $\\mu(u) = u$ \nand $\\mu(v) = v-u^{k-1}$. For the second part, define $\\mu : K X \\rightarrow \nK J_{n-1}$ via $\\mu(u) = u$ and $\\mu(v) = u^{n\/2-1} - \\sqrt{-1} v$. Then \nas above, it follows that $\\mu$ extends to an isomorphism. \n\\medskip\n\nShow that $K N_1 \\cong K N_2$ and $K N_3 \\cong K N_4$ hold.\\\\\nNote that $(N_1,N_2)$ and $(N_3,N_4)$ are pairs of \nanti-isomorphic semigroups. For each $i \\in \\{1, \\ldots, 4\\}$, the \nsubsemigroup $\\langle u, u^{n-3}-v\\rangle$ yields a basis of $K N_i$ \nand is isomorphic to the dual semigroup of $N_i$. Hence $K N_1 \\cong \nK N_2$ and $K N_3 \\cong K N_4$ follow.\n\\medskip\n\nIt remains to show that we have determined all isomorphisms. First, \nwe consider $K H_{n-1}$ and $K J_{n-1}$. These are both \ncommutative algebras; the first has an annihilator of dimension 2\ngenerated by $v$ and $u^{n-2}$ and the second has an annihilator of\ndimension 1 generated by $u^{n-2}$. Hence the algebras are non-isomorphic.\nSecondly, we consider  $K N_1$ and $K N_3$. These are both\nnon-commutative algebras and they both have an annihilator of dimension 1; \nthe first has a right annihilator of dimension 2 generated by $v$ \nand $u^{n-2}$ and the second has a right annihilator of dimension 1\ngenerated by $u^{n-2}$. Hence the algebras are non-isomorphic. This\nproves our claim in the case $\\sqrt{-1} \\in K$ or $n$ odd. In the case \n$\\sqrt{-1} \\not \\in K$ and $n$ even, there is the additional algebra \n$K X$. This is a commutative algebra whose annihilator has dimension 1;\nhence we have to distinguish $K X$ from $K J_{n-1}$. Assume that $\\mu : \nK X \\rightarrow K J_{n-1}$ is an isomorphism and denote $\\mu(v) = av + \n\\sum_{i=1}^{n-2} b_i u^i \\in K J_{n-1}$. Then $\\mu(v)^2 = a^2 v^2+\n(\\sum_{i=1}^{n-2} b_i u^i)^2$ in $K J_{n-1}$, as $uv = vu = 0$ holds. \nNote that $v^2 = u^{n-2}$ in $K J_{n-1}$ and $\\mu(v)^2 = \\mu(v^2) = \n0\\in K J_{n-1}$ as $v^2=0$ in $KX$. \nAn inspection of the coefficients now shows that $b_i = 0$ for $1 \\leq i \n\\leq n\/2-2$. The coefficient of $u^{n-2}$ in $\\mu(v)^2$ thus is \n$a^2+b_{n\/2-1}^2$. Since $\\sqrt{-1} \\not \\in K$, it follows that \n$a = b_{n\/2-1} = 0$. This yields that $\\mu(v) \\in \\langle u^{n\/2}, \nu^{n\/2+1}, \\ldots, u_{n-2} \\rangle \\leq (K J_{n-1})^2$. Hence $\\mu$ \nis not surjective, a contradiction.\n\\medskip\n\nWe determine the edges of $\\mathcal G_{1,K}$. Consider a semigroup $S$ of order\n$n$ from the above classification. In the quotient $S\/S^{n-2}$ the two\nelements $u^{n-2}$ and $u^{n-1}$ are identified, and hence the quotient \nis isomorphic to a semigroup of type $H_k$ of order $n-1$. Note that the \nlater is valid for $n=5$ also, as the semigroups $H_k$ can be defined for \norder $4$ as well.\n\\medskip\n\nThe labels of the vertices in $\\mathcal G_{1,K}$ follow immediately from the\nclassification. This implies that $\\mathcal G_{1,K}$ has strong defect 2 and\nstrong period 2. (In fact, both values are minimal.)\n\\end{proof}\n\nImages of the parts of $\\mathcal G_{1,K,2}$ corresponding to semigroups\nof order at most $12$ for $K = GF(p)$ with $p \\leq 23$ can be found\nat~\\cite{homepage}.\n \n\\section{Computational experiments for $\\mathcal G_{2,K}$}\n\\label{cc2}\n\nA classification of semigroups of coclass $2$ is available in\n\\cite{Dis10,Dis11}. We\nused it to investigate $\\mathcal G_{2,K}$ computationally, applying the\nisomorphism test for associative nilpotent algebras over finite fields\nin~\\cite{Eic07}. Semigroups of coclass $2$ have a minimal\ngenerating set of size $2$ or $3$. We know from Section~\\ref{sec_mingen}\nthat these two cases lead to independent subgraphs $\\mathcal G_{2,K,2}$ and\n$\\mathcal G_{2,K,3}$ of $\\mathcal G_{2,K}$ which shall be considered separately.\n\nWe have determined the part of $\\mathcal G_{2,K,2}$ corresponding to semigroups\nof order at most $12$ for $K = GF(p)$ with $p \\leq 23$. Images of the\ngraphs are available at~\\cite{homepage}. We conjecture\nthat for every field $K$ the graph $\\mathcal G_{2,K,2}$ has five maximal infinite\npaths which are described by the following infinite dimensional\nalgebras:\n\\begin{items}\n\\item[$\\bullet$]\n$\\langle a, b \\mid b^2 = ba = a^2b = 0 \\rangle$\nwith annihilator $\\langle ab \\rangle$;\n\\item[$\\bullet$]\n$\\langle a, b \\mid b^2 = ab = ba^2 = 0 \\rangle$\nwith annihilator $\\langle ba \\rangle$;\n\\item[$\\bullet$]\n$\\langle a, b \\mid b^3 = ab = ba = 0 \\rangle$\nwith annihilator $\\langle b^2 \\rangle$;\n\\item[$\\bullet$]\n$\\langle a, b \\mid b^2 = aba = 0, ab = ba \\rangle$\nwith annihilator $\\langle ba \\rangle$;\n\\item[$\\bullet$]\n$\\langle a, b \\mid b^2 = ba, ab = b^2a = 0 \\rangle$\nwith annihilator $\\langle ba \\rangle$.\n\\end{items}\nUsing these presentations to define semigroups with zero we obtain\ninfinite semigroups that are annihilator extensions of the semigroup\nunderlying $M_{K,1}$ and whose contracted semigroup algebras are the\nalgebras defined by the presentations. \nIf the conjecture on the number of infinite paths holds, then\n$\\mathcal G_{2,K,2}$ contains five maximal coclass\ntrees. Figure~\\ref{figcc2d2} exhibits the respective trees of the\ncomputed graph for $K=GF(5)$.\nFurthermore our computational evidence suggests the following:\n\n\\begin{items}\n\\item[$\\bullet$]\nthe graph $\\mathcal G_{2,GF(p),2}$ depends on $p \\bmod 4$ only;\n\\item[$\\bullet$]\nthe vertices in $\\mathcal G_{2,GF(p),2}$ outside a maximal coclass tree have \ndimension $4$ or $5$;\n\\item[$\\bullet$]\nthe roots of the maximal coclass trees of $\\mathcal G_{2,GF(p),2}$ have dimension $4$;\n\\item[$\\bullet$]\neach maximal coclass tree in $\\mathcal G_{2,GF(p),2}$ is strongly virtually\nperiodic; one tree has strong defect $1$ and strong period $1$, the\nother four trees have strong defect $2$ and strong period $2$;\n\\item[$\\bullet$]\nthe arising strong defects and strong periods are independent of the field.\n\\end{items}\n\nAdditionally the polynomials describing the labels in the\nperiodic parts of the maximal coclass trees have degree at most $1$, a\nfact that follows from the classification in~\\cite{Dis10,Dis11}.\n\n\\begin{figure}[thb]\n\\begin{center}\n\\includegraphics[scale=0.4]{cc2.png}\n\\end{center}\n\\vspace{-1cm}\n\\caption{Maximal coclass trees in $\\mathcal G_{2,GF(5),2}$ up to depth 12.}\n\\label{figcc2d2}\n\\end{figure}\n\nThe graph $\\mathcal G_{2,K,3}$ is known to consist of a single maximal coclass\ntree with root $KZ_4$ and infinite paths corresponding to $M_{K,2}$ by\nTheorem~\\ref{mingen}. We have determined the part of $\\mathcal G_{2,K,3}$\ncorresponding to semigroups of order at most $12$ for $K = GF(p)$ with\n$p \\leq 5$. Images of the graphs are available at~\\cite{homepage}.\nIn all three cases we found the graph to appear strongly\nvirtually periodic with strong defect $2$, and strong period $2$. In\naccordance with the results from~\\cite{Dis10,Dis11} the labels can be\ndescribed by quadratic polynomials.\n\n\\section{Computational experiments for $\\mathcal G_{3,K}$}\n\\label{cc3}\n\nFor the semigroups of coclass $3$ there is no general classification known. \nWe computed the semigroups of coclass $3$ and order at most $17$ up to \nisomorphism using the code provided in~\\cite[Appendix C]{Dis10}. Then we\ndetermined the part of $\\mathcal G_{3,K,2}$ correspondig to these semigroups for \n$K = GF(p)$ with $p \\leq 23$. Images of the graphs are available \nat~\\cite{homepage}. We summarise our observations:\n\\begin{items}\n\\item[$\\bullet$]\nthe graph $\\mathcal G_{3,GF(p),2}$ depends on $p \\bmod 4$ only;\n\\item[$\\bullet$]\nthe graph $\\mathcal G_{3,GF(p),2}$ has $15$ maximal infinite paths of which\n$4$ correspond to commutative algebras; \n\\item[$\\bullet$]\nthe vertices in $\\mathcal G_{3,GF(p),2}$ outside a maximal coclass tree have \ndimension $6$, $7$ or $8$;\n\\item[$\\bullet$]\nthe roots of the maximal coclass trees of $\\mathcal G_{3,GF(p),2}$ have\ndimension $5$, $6$ or $7$;\n\\item[$\\bullet$]\nthe maximal coclass trees in $\\mathcal G_{3,GF(p),2}$ are strongly virtually\nperiodic with strong defect at most $3$ and strong period at most $6$;\n\\item[$\\bullet$]\nthe arising strong periods are independent of the field;\n\\item[$\\bullet$]\nthe polynomials describing the labels have degree at most $1$.\n\\end{items}\n\nThese observations for $\\mathcal G_{3,GF(p),2}$ are of particular\ninterest as this is the first case in which some of the semigroups\ncontain products of three elements that lie in different monogenic\nsubsemigroups. In fact, $\\mathcal G_{3,GF(p),2}$ has in many aspects more \ncomplex features than all other considered graphs.\n\nWe have investigated the part of $\\mathcal G_{3,K,3}$ correspondig to semigroups of\norder at most $12$ for $K = GF(2)$ only. There appear to be $21$\nmaximal infinite paths in $\\mathcal G_{3,GF(2),3}$ with $5$ paths\ncorresponding to commutative algebras. \n\nThe graph $\\mathcal G_{3,K,4}$ has $1$ maximal infinite paths corresponding\nto the commutative algebra $M_{K,3}$ by Theorem~\\ref{mingen}.\n\n\\section{Concluding comments}\n\\label{final}\n\nSimilar to the graphs $\\mathcal G_{r,K}$ one can define a graph $\\mathcal G_r$ whose \nvertices correspond one-to-one to the isomorphism types of semigroups \nof coclass $r$. Two vertices are adjoined by a directed edge $T \\rightarrow S$\nif $S\/S^c \\cong T$ where $c$ is the class of $S$.\nIt follows directly from \\cite[Lemma 3.1]{Dis11} that the graph $\\mathcal G_r$\ndoes not have finite width (unless $r=0$).\n\nThe additional use of the contracted semigroup algebras in the definition\nof $\\mathcal G_{r,K}$ induces a dependence on the underlying field $K$, but it has \nthe significant advantage that the graphs $\\mathcal G_{r,K}$ seem to have finite \nwidth and exhibit periodic patterns which can be described in a compact way. \nFurther, the considered field $K$ seems to have no influence on the important\naspects of the periodicity. \n\n\\def$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\n\nThe monitoring of storage, movement, and quality of water at regional and global scales is of vital importance to practical applications such as agricultural production, water resources management, and predictions of flood, drought and climate change \\citep{Seneviratne2010, Wood2011, Zink2016}. To study land surface processes, soil moisture information have to be included, averaged at a scale relevant and representative for the physical, chemical, and biological processes \\citep{Entekhabi1999, Corwin2006, Schulz2006, Gentine2012, Vereecken2015}. The provision of parameters describing the critical processes at the landscape scale and capturing the natural heterogeneity of the soil-hydrological system at scales of 1 to 1000 m is one of the grand challenges in soil moisture monitoring \\citep{Robinson2008, PetersLidard2017}.\n\nOver the last 10--15 years, satellite-based Earth observation technologies made enormous progress. The potential of satellite-based remote sensing to map soil moisture dynamics at the catchment scale (\\(\\sim1000\\,\\mathrm{km^2}\\)) has been demonstrated by numerous studies \\citep{Kerr2007, Famiglietti2008, Wagner2009, WangQu2009, Liu2011, Ochsner2013}. While such data are widely used today to calibrate large-scale hydrological models up to the global scale, satellite-based remotely sensed soil moisture information are often not appropriate to reveal processes at the intermediate scale (up to 1000 m) \\citep{Western2002}. The reasons for this are, e.g., the shallow measurement depth, disturbing influences of vegetation or surface roughness on the signal and resulting lacks in the data quality, and a too coarse spatial resolution \\citep{Robinson2008}. The accuracy and precision of remotely sensed products is also not constant around the globe, which is less an issue for ground-based methods. Comparing the spatiotemporal coverage of remotely sensed soil moisture against spatiotemporal scales covered by local instruments (e.g.~TDR, gravimetry, EMI\/ERT, gamma-rays, NMR), it becomes obvious that ``there is currently a gap in our ability to routinely measure at intermediate scales'' \\citep{Robinson2008}.\n\nThe method of Cosmic-Ray Neutron Sensing (CRNS) for soil moisture estimation, introduced to the environmental science community by \\citet{Zreda2008}, provides a much larger measurement footprint than any other ground-based local method. With a support volume in the order of \\(10^4\\) m\\(^3\\) (\\(>100\\,\\)m radius, \\(<0.8\\,\\)m depth, \\citet{Koehli2015}), CRNS has a large potential to close the scale gap between point measurements of root-zone soil moisture and remotely sensed surface soil moisture \\citep{Ochsner2013, Montzka2017}. The CRNS technology makes use of the extraordinary high sensitivity of cosmic-ray neutrons to hydrogen nuclei and measures the concentration of epithermal neutrons above the soil surface. Since its introduction, the CRNS technology has quickly established itself and is now used for soil moisture monitoring by many research groups operating worldwide \\citep[e.g.,][]{Bogena2013, Franz2013, Peterson2016, Zhu2016, Schroen2017w}.\n\nPilot studies have shown the concept and potential of \\emph{mobile} CRNS \\citep{Desilets2010} using neutron detectors mounted on a ground-based vehicle (``rover''). The method is comparable to exploration missions with rovers on the Martian surface \\citep{Jun2013}. With advances on understanding the CRNS method also for stationary probes, recent studies have more and more elaborated on direct applications of the so-called \\emph{CRNS rover} \\citep{Chrisman2013, McJannet2014, Dong2014, Franz2015, Avery2016}. While the ``classical'', stationary CRNS application enables to capture the hourly variability of soil moisture within a static footprint, the mobile application is intended to capture the spatial variability of soil moisture across larger areas or along larger transects. The CRNS rover uses the same detection principle as the stationary CRNS probes but deploys multiple and larger neutron detectors in order to achieve higher count rates at much shorter recording periods.\n\nAgricultural fields and private land are often not accessible by vehicles. Hence, the CRNS rover is usually moved along a network of existing roads, streets, and pathways in a study region. This strategy is also practical when the rover is used to cover large areas at the regional scale in a short period of time. However, recent findings by \\citet{Koehli2015} have shown that the CRNS detector is particularly sensitive to the first few meters around the sensor, which was later confirmed by calibration and validation campaigns of stationary CRNS probes \\citep{Schroen2017w, Heidbuechel2016, Schattan2017}. This aspect is of even higher importance for the mobile application of CRNS. Following this argumentation, we hypothesize that the CRNS measurement is biased significantly when the moisture conditions present in the road differ substantially from the actual field of interest.\n\nThe effect of dry structures in the footprint was introduced for the first time by \\citet{Franz2013} and was observed by \\citet{Chrisman2013} on rover surveys through urban areas. \\citet{Franz2015} sensed soil moisture of agricultural fields by roving on paved and gravel roads, and speculated that the road material could have introduced a dry bias to their measurements. The quantification of this effect is critical, not only for the advancement of the method, but also for its application to support agricultural irrigation management \\citep{Franz2015} or to allow for large-scale soil moisture retrieval to support hydrological modeling \\citep{Zink2016, Schroen2017diss} and evaluation of remote-sensing products \\citep{Montzka2017}.\n\nIn the present study, we aim to evaluate and quantify the ``road effect'' by combining physical neutron transport modeling and dedicated field experiments. Based on theoretical investigations, we propose a universal correction function which is then tested and discussed in the light of ten rover campaigns in Central Germany and South England.\n\n\\section{Methods}\\label{methods}\n\n\\subsection{The Cosmic-Ray Neutron Rover}\\label{the-cosmic-ray-neutron-rover}\n\nThe method of cosmic-ray neutron sensing makes use of thermal neutron detectors filled with helium-3 or borontrifluorid \\citep{Persons2011, Schroen2017u}. A surrounding shield of polyethylene prevents most thermal neutrons in the natural radiation environment from entering the detector, while it slows down incoming, epithermal neutrons to detectable, thermal energies \\citep{Zreda2012, Andreasen2016}. The epithermal neutron density in air is mainly controlled by (1) the interaction of direct cosmic radiation with the ground, and (2) the number of hydrogen atoms in the environment \\citep{Zreda2008, Koehli2015}. As hydrogen is an elemental part of the water molecule, the correlation between the epithermal neutron signal and surrounding water storages can be beneficial for the monitoring of the hydrological cycle. Figure \\ref{fig:maps} shows a combination of the helium-3 detector system (white case, left), a small helium-3 unit (black case, middle), and four borontrifluorid tubes (right) which have been disassembled from stationary probes.\n\nThe mobile CRNS detectors can be mounted in the trunk of a car. As neutrons are almost exclusively sensitive to hydrogen, the metallic material of the car appears almost transparent. Additional plastic components and human presence can have the effect of a constant shielding factor, which is irrelevant for CRNS applications as only relative changes of neutrons are evaluated. Air temperature and humidity are recorded with sensors mounted externally to the car, because air conditions inside and outside can differ significantly. The neutron detector was set to integrate neutron counts over 1 minute. When in motion, this implicitly stretches the otherwise circular footprint to a patch elongated in the driving direction. In contrast, the GPS coordinates are read from a \\texttt{Globalsat\\ BR-355} sensor at the time of recording, so after the neutrons were integrated. To account for this artificial shift in post-processing mode, the UTM coordinates of each signal were back-projected to half of the distance covered within that minute. Driving speed was adapted on local structures and ranged from 15 to \\(80\\) meters per minute.\nThe neutron count rate \\(N\\) depends on environmental moisture conditions and on the detector volume used. The helium-3 detector system observed \\(90\\) to \\(170\\) counts per minute (cpm) and showed a similar count rate as the sum of four borontrifluorid tubes. Three consecutive measurements (i.e., 3 minutes) underwent a moving-average filter to account for the moving footprint and to reduce the relative statistical uncertainty, \\(\\sqrt{N}\/N\\), by a factor of \\(\\sqrt{3}\\approx1.73\\).\n\nBesides near-surface water content, neutron radiation in the environment mainly depends on the incoming variation of cosmic rays, on the air mass above the sensor (and thus on altitude), and on water vapor in the air \\citep{Schroen2015}. In this work, we have applied standard procedures to correct for these effects \\citep{Zreda2012, Rosolem2013, Hawdon2014} in order to obtain a processed neutron count rate \\(N\\).\n\nTo convert the neutron count rate to gravimetric soil water equivalent, \\(\\theta_\\text{grv}\\), several approaches have been proposed in literature. \\citet{Desilets2010} suggested a theoretical relation that has been applied successfully by the majority of CRNS studies in the past. \\citet{McJannet2014} found that this approach performs also better for rover campaigns than the universal calibration function proposed by \\citet{Franz2013u}, as the exact determination of soil and land-use data is the major obstacle to apply the latter. The standard approach from \\citet{Desilets2010} is as follows:\n\\begin{linenomath*}\n\\begin{equation} \\theta_\\text{grv}(N,N_0) = \\frac{a_0}{N\/N_0-a_1}-a_2\\,, \\label{eq:desilets}\\end{equation}\n\\end{linenomath*}\nwhere parameters \\(a_i=\\{0.0808, 0.372, 0.115\\}\\) were determined using neutron physics simulations, and \\(N_0\\) is a (site-specific) calibration parameter. The latter is determined once for each dataset by comparing the CRNS soil moisture product with the actual soil moisture condition in the field. However, neutrons are sensitive to all occurances of hydrogen in the footprint, such as ponds, organic material, lattice water, plant water, and other dynamic contributors. Hence, the variable \\(\\theta_\\text{grv}(N,N_0)=\\theta_\\text{sm}+\\theta_\\text{offset}\\) denotes the sum of the soil water equivalent and an offset introduced by additional hydrogen pools. Furthermore, to compare CRNS products with other point sensors, the gravimetric water content is converted to volumetric water content, \\(\\theta_\\text{vol}=\\theta_\\text{grv}\\cdot\\varrho_\\text{bd}\\), using soil bulk density information \\(\\varrho_\\text{bd}\\). In this work, we define\n\\begin{linenomath*}\n\\begin{equation} \\theta(N)=\\varrho_\\text{bd}\\,(\\theta_\\text{grv}(N,N_0)-\\theta_\\text{offset}) \\label{eq:roversm}\\end{equation}\n\\end{linenomath*}\nas the CRNS soil moisture product, given in units of volumetric percent (\\%) throughout this manuscript.\n\nTo account for spatially variable parameters of soil bulk density and land use throughout the study area, additional sources of data have been incorporated by recent studies \\citep{Avery2016, Schroen2017diss, McJannet2017}. However, spatial information at the field scale (1--100 m) is often not available or come with significant uncertainty. This can be considered as a general handicap of the mobile CRNS method. In this work, we decided to apply the standard approach using spatially constant parameters, because (1) the selected study sites show sufficiently homogeneous soil and land-use conditions, and (2) the focus of the present study is to quantify the local effect of roads to the relative neutron signal, rather than the exact estimation of absolute soil moisture.\n\n\\subsection{Validation with point-scale measurements}\\label{sec:weight}\n\nSince the footprint of the CRNS signal covers an area of several hectares, comparison with point data is a challenge. To bridge this scale gap, \\citet{Schroen2017w} developed a procedure to calculate a weighted average of point samples, based on their distance and depth to the neutron detector. The method uses an advanced spatial sensitivity function based on neutron transport simulations by \\citet{Koehli2015}, and was successfully applied to calibration and validation datasets for stationary CRNS probes.\n\nIn our work presented here, we employed independent validation measurements of field soil moisture in the first 10 centimeters using occasional soil samples, and high frequency electromagnetic measurements with \\texttt{Campbell\\ TDR\\ 100} and \\texttt{Theta\\ Probes}. The latter both instruments are standard approaches to determine near-surface soil water \\citep{Roth1990}. The \\texttt{Theta\\ Probe} measures soil system impedance at \\(100\\,\\)MHz, while the \\texttt{TDR\\ 100} evaluates pulse travel time in the GHz-range \\citep[see also][]{Blonquist2005, Robinson2003, Vaz2013}.\n\nIn order to compare the point measurements with the CRNS soil moisture product, a weighted average of the point data is applied based on their individual distance \\(r\\) to the neutron detector (see illustration circle in Fig.~\\ref{fig:maps}). Using eqs. \\ref{eq:desilets}--\\ref{eq:roversm}, the calibration parameter \\(N_0\\) can be determined from the neutron count rate \\(N\\) and the independently measured value for average field soil moisture, \\(\\langle\\theta\\rangle\\).\nThe soil moisture products have been interpolated using an \\emph{Ordinary Kriging} approach, as the chosen measurement density adequately represents typical spatial correlation lengths of soil moisture at our study sites. Furthermore, the Kriging approach supports the non-local nature of the epithermal neutron distribution in the air \\citep{Franz2015, DesiletsZreda2013, Koehli2015}.\n\n\n\\subsection{Experimental setup}\\label{sec:sites}\n\n\\subsubsection{Road types}\\label{road-types}\n\nRoad moisture content is typically an uncertain quantity, only accessible by destructive sampling and lab analysis, or expensive geophysical exploration \\citep{Saarenketo2000, Benedetto2012}. In the scope of the uncertainties involved in neutron sensing, e.g.~due to spatial heterogeneity of roads and surrounding land use, visual determination of the road material, guided by literature information, can allow for an adequate estimate of its elemental composition and thus, its soil water equivalent. \\citet{Chrisman2013} analyzed several samples of stone\/concrete and asphalt in Arizona and found their \\emph{gravimetric} water equivalent to be \\(1.52\\,\\%\\) and \\(5.10\\,\\%\\), respectively. Following literature values for typical material densities from \\(1.8\\,\\mathrm{g\/cm^3}\\) (sandy concrete) to \\(2.4\\,\\mathrm{g\/cm^3}\\) (hot asphalt) \\citep{Houben1994, Stroup2000}, the \\emph{volumetric} water equivalent then is \\(\\approx3\\,\\%\\) and \\(\\approx12\\,\\%\\), respectively. As stone and asphalt are known as one of the most ``dry'' and ``wet'' road materials, respectively, we have estimated the moisture content of the various road types in our study sites within this range of extremes.\n\n\\subsubsection{Sch{\\\"a}fertal (Germany, experiment A)}\\label{schuxe4fertal-germany-experiment-a}\n\nThe \\emph{Sch{\\\"a}fertal} site is a headwater catchment in the \\emph{Lower Harz Mountains} and one of the intensive monitoring sites in the \\emph{TERENO Harz\/Central German Lowland Observatory} (\\(51^\\circ\\,39'\\,\\)N, \\(11^\\circ\\,3'\\,\\)E) \\citep{Zacharias2011, Wollschlaeger2016}. The catchment covers an area of 1.66 km\\(^2\\) and is predominantly under agricultural management. At the valley bottom, grassland surrounds the course of the creek \\emph{Sch{\\\"a}ferbach}. Grassland is also present at the outlet of the catchment and a forest occupies a small area at the north-eastern end of the catchment outlet. Average climatology shows mean annual minimum and maximum temperatures of \\(-1.8^\\circ\\)C and \\(15.5^\\circ\\)C, respectively; and mean annual precipitation of 630 mm.\nAverage bulk density of the soil is \\(\\langle\\varrho_\\text{bd}\\rangle=1.55\\,\\mathrm{g\/cm^3}\\) and water equivalent of additional hydrogen and organic pools have been approximated to be \\(\\langle\\theta_\\text{offset}\\rangle=2.3\\,\\%\\) in bare soil. For more information about the local hydrology, see \\citet{Martini2015} and \\citet{Schroeter2015}.\n\nWithin the Sch{\\\"a}fertal, \\citet{Schroeter2015, Schroeter2017} performed regular TDR campaigns by foot using 94 locations in the whole catchment area. During several campaigns from 2014--2016, the CRNS rover accompanied their team. Shortly after harvest the fields were accessible with the car, such that the same locations could be sampled with the rover and the TDR team on the same campaign day.\nOn some days, however, the vehicle was only allowed to access the fields due to agricultural activities and seeded vegetation, such that CRNS measurements were taken on the sandy roads that were crossing the agricultural fields and the creek.\n\nThe road network consists mainly of three types: a paved major road between the hilltops and the urban area, sandy roads within the catchment, and pathways along the creek. The paved road has an average width of \\(w\\approx3.5\\,\\)m and consists of a very dry stone\/concrete mixture that was estimated to contain \\(\\theta_\\text{road}\\approx4\\,\\%\\) volumetric water content. The secondary roads are a mixture of stone, sand, and gravel, with \\(\\theta_\\text{road}\\approx6\\,\\%\\) and width \\(w\\approx3\\,\\)m. The pathways of width \\(w\\approx3\\,\\)m contain mixed material from gravel, soil, and grass with an estimated average moisture content of \\(\\theta_\\text{road}\\approx10\\,\\%\\).\n\nRover measurements have been performed using the helium-3 detector system at count rates of approximately \\(90\\) to \\(170\\,\\)cpm depending on wetness conditions. The corresponding neutron count uncertainties of \\(6\\) to \\(4\\,\\%\\) propagated through eq.~\\ref{eq:desilets} to absolute uncertainties in water equivalent, \\(\\Delta\\theta_\\text{grv}\\), of \\(10.0\\) to \\(0.9\\) gravimetric percent, for wet to dry conditions, respectively.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics{maps.pdf}\n\\caption{The study sites \\emph{Sch{\\\"a}fertal} (top right) and \\emph{Sheepdrove Organic Farm} (bottom right). White borders indicate the areas of three different field experiments A, B, and C. Black lines indicate the type of road. The central circle with TDR points (red) and rover points (blue) illustrates the spatial calibration of the CRNS rover by comparing the large-scale neutron counts with point-scale soil moisture, using a weighted average of point samples based on their distance \\(r\\) to the rover. Pictures at certain spots: Sch{\\\"a}fertal gravel\/sand road (P1), Sheepdrove valley (P2), Sheepdrove gravel\/stone road, asphalt\/stone road close-up (P4), and neutron detector tubes in the trunk of a car which is used as a CRNS rover.}\\label{fig:maps}\n\\end{figure}\n\n\n\\subsubsection{Sheepdrove Organic Farm (England, experiments B and C)}\\label{sheepdrove-organic-farm-england-experiments-b-and-c}\n\nThe \\emph{Sheepdrove Organic Farm} is located on the \\emph{West Berkshire Downs} in the Lambourn catchment in South England (\\(51^\\circ\\,32'\\,\\)N, \\(1^\\circ\\,29'\\,\\)W). The farm is located in a dry valley with elevations ranging from \\(140\\,\\)m to \\(200\\,\\)m above Ordnance Datum. The hydrogeology is characterized by a highly permeable white chalk aquifer with the groundwater table located tens of meters below the surface \\citep{Evans2016}. Average climatology obtained from the Marlborough meteorological station (located 22 km to the south-west from the farm) shows mean annual minimum and maximum temperatures of \\(5.4^\\circ\\)C and \\(14.0^\\circ\\)C, respectively; and mean annual precipitation of 815 mm.\nSoil information at the farm was collected at three sites with slightly different soil\/vegetation characteristics between 2015 and 2017 \\citep{Iwema2017}. The soil is generally loamy clay with many flints and pieces of chalk. Weathered chalk starts below the soil at about 25 to 40 centimeters depth. The average bulk density is \\(\\langle\\varrho_\\text{bd}\\rangle=1.25\\,\\mathrm{g\/cm^3}\\) and water equivalent of additional hydrogen and organic pools have been determined to be \\(\\langle\\theta_\\text{offset}\\rangle=4.3\\,\\%\\) using soil sample analysis.\n\nThe road network consists of a paved major road (width \\(w\\approx3\\,\\)m) made of an asphalt\/stone mixture with an estimated moisture content of \\(\\theta_\\text{road}\\approx11\\,\\%\\). The main side roads are made of a gravel\/stone mixture (\\(\\theta_\\text{road}\\approx7\\,\\%\\)), most of which are \\(w\\approx2.3\\,\\)m wide while the southern road is \\(w\\approx4.5\\,\\)m wide. Many non-sealed tracks (\\(w\\approx3\\,\\)m) follow the borders between fields which partly consist of sand, grass, and organic material, such that their average moisture content was estimated to \\(\\theta_\\text{road}\\approx12\\,\\%\\).\n\nRover measurements have been performed using the combination of the helium-3 detector system and the four borontrifluorid tubes at total count rates of approximately \\(180\\) to \\(330\\,\\)cpm depending on wetness conditions. The corresponding neutron count uncertainties of \\(4\\) to \\(3\\,\\%\\) propagated through eq.~\\ref{eq:desilets} to absolute uncertainties in water equivalent, \\(\\Delta\\theta_\\text{grv}\\), of \\(7.5\\) to \\(0.6\\) gravimetric percent, for wet to dry conditions, respectively.\n\n\\subsection{Simulation of neutron interactions with road structures}\\label{sec:theory}\n\nTheoretical calculations of the CRNS footprint by \\citet{Koehli2015} have shown that the radial sensitivity of a CRNS detector is strongly pronounced in the first few meters around the sensor \\citep[see also][]{Schroen2017w}. Therefore, this work hypothesizes that there is an influence from the nearby road material to the neutron signal \\(N\\), which differs from the signal \\(N_\\text{field}\\) measured above the soil if no roads were present. In this regard, we define the bias \\(N\/N_\\text{field}\\neq 1\\) describing the relative deviation of measured neutrons \\(N\\) on the road from measurements on the field, \\(N_\\text{field}\\), if the moisture contents of road and soil differ.\n\nMany mobile surveys rely on road-only measurements of cosmic-ray neutrons, but we can expect that a potential road effect is larger for larger differences between road moisture and surrounding field water content. It is highly impractical to measure the corresponding bias rigorously, as it might depend also on the road material (see above), on field soil moisture, and on the distance to the road. We therefore employed the Monte Carlo technique using the neutron transport code \\texttt{URANOS} (\\citet{Koehli2015}, www.ufz.de\/uranos) to simulate neutron response in a domain of 25 hectares which is crossed by a straight road geometry (see Fig.~\\ref{fig:uranos}).\n\nThe road is modeled as a \\(20\\,\\)cm deep layer of either stone or asphalt, while the soil below was set to \\(5\\,\\%\\) volumetric water content.\nFollowing the compendium of material composition data \\citep{McConn2011}, asphalt pavement is modeled as a mixture of O, H, C, and Si, at an effective density of \\(\\mathrm{2.58\\,g\/cm^3}\\), which corresponds to a soil water equivalent of \\(\\approx12\\,\\%\\). Stone\/gravel is a mixture of Si, O, and Al, plus \\(3\\,\\%\\) volumetric water content at a total density of \\(\\mathrm{1.4\\,g\/cm^3}\\) \\citep{Koehli2015}.\nThe wetness of the surrounding soil has been set homogeneously to 10, 20, 30, and 40\\(\\,\\%\\) of volumetric water content. The neutron response to roads has been simulated for road widths of 3, 5, and 7\\(\\,\\)m.\n\n\\clearpage\n\n\\begin{figure}\n\\centering\n\\includegraphics{uranos-schematic.pdf}\n\\caption{(a) Schematic of the model setup used by the Monte-Carlo code \\texttt{URANOS} to simulate the response of cosmic-ray neutrons to ground materials. (b) Exemplary \\texttt{URANOS} model output showing a birds-eye view of the neutron density in the horizontal detector layer.}\\label{fig:uranos}\n\\end{figure}\n\n\n\\section{Results \\& Discussions}\\label{results-discussions}\n\n\\subsection{Theoretical investigations}\\label{theoretical-investigations}\n\nThe spatial Monte-Carlo simulations have been performed to study the interactions of cosmic-ray neutrons with roads of various widths, materials, and homogeneous field soil moisture conditions. The term ``relative road bias'' denotes the ratio of neutron intensity \\(N\\) detected in a road scenario (see Fig.~\\ref{fig:uranos}) over neutron intensity \\(N_\\text{field}\\) detected in a scenario of homogeneous soil moisture.\n\nSymbols in Fig.~\\ref{fig:croadsm} show the simulated road bias for a detector placed at the center of the road. The bias increases for increasing field soil moisture, increasing road width, and decreasing road moisture. The quantity is particularly sensitive to the water equivalent of the pavement (\\(\\theta_\\text{road}\\)) and the soil (\\(\\theta_\\text{field}\\)).\nFigure \\ref{fig:croadr} plots the simulated road bias over distance from the road center, showing that the bias is a short-range effect that decreases a few meters away from the road, where almost no measurable effect can be expected beyond \\(\\approx10\\,\\)m distance. It is evident from these simulations that the road bias is higher the larger the difference between road moisture and surrounding soil moisture is and the wider the road.\n\nWe suggest to correct the observed neutron intensity with a correction factor \\(C_\\text{road}\\), similar to the approaches used to correct for meteorological \\citep{Hawdon2014, Schroen2015} and biomass effects \\citep{Baatz2014}:\n\\begin{linenomath*}\n\\begin{equation}\nN_\\text{corr} = N \/ C_\\text{road}\\,,\n\\label{eq:croadapproach}\\end{equation}\n\\end{linenomath*}\nwhere the correction factor should be 1 for no-road conditions, plus a product of terms that depend on the characteristics of the road and field conditions. The shape of each term of the proposed correction function is based on physical reasoning as follows:\n\n\\begin{enumerate}\n\\def\\arabic{enumi}.{\\arabic{enumi}.}\n\n\\item\n  The dependence on road width \\(w\\) is assumed to be a simple exponential, since the short-range dependency of neutron intensity on distance is exponential as shown by \\citet{Koehli2015}.\n\\item\n  The dependence on water content (\\(\\theta_\\text{road}\\) and \\(\\theta_\\text{field}\\)) is assumed to be hyperbolic, since the natural response of neutrons to soil water exhibits hyperbolic shape, as was derived from basic principles by \\citet{Desilets2010} and \\citet{Schroen2017diss}. This form (e.g., eq.~\\ref{eq:desilets}) has been proven to be robust among all studies related to CRNS so far.\n\\item\n  The dependence on distance \\(r\\) from the road center is assumed to be a sum of exponentials, since the combination of short- and long-range neutrons indicate this picture \\citep[see][]{Koehli2015}. An additional polynom (\\(w^a\\, r^b\\)) might be necessary to account for the plateau introduced by the road of a certain width \\(w\\).\n\\item\n  Additionally, we demand that the total correction factor is 1 for road widths \\(w=0\\) and for similar moisture conditions, \\(\\theta_\\text{road}=\\theta_\\text{field}\\). The dependency on distance should be further normalized to 1 at the road center (\\(r=0\\)).\n\\end{enumerate}\n\nThe semi-analytical approach has been fitted to the \\texttt{URANOS} simulation results. A minimum of 11 numerical parameters were required in order to adequately capture the most prominent features and dependencies of the simulated neutron response:\n\n\\begin{linenomath*}\n\\begin{equation}\nC_\\text{road}(\\theta_\\text{field}, \\theta_\\text{road}, w, r)\n    = 1+F_1(w)\\cdot F_2(\\theta_\\text{field},\n    \\theta_\\text{road})\\cdot F_3(r, w)\\,,\n\\label{eq:croad}\\end{equation}\n\\end{linenomath*}\nwhere\n\\begin{linenomath*}\n\\begin{equation}\n\\begin{aligned}\nF_1(w)\n    &= p_0\\,\\big(1-e^{-p_1\\,w}\\big)\\,,\\\\\nF_2(\\theta_\\text{field}, \\theta_\\text{road})\n    &= \\big(\\theta_\\text{field}-\\theta_\\text{road}\\big)\\frac{p_2-p_3\\,\\theta_\\text{road}}{\\theta_\\text{field}-p_4\\,\\theta_\\text{road}+p_5}\\,,\\\\\nF_3(r, w)\n    &= p_6\\,e^{-p_7\\,w^{-p_8}\\,r^4}+p_9\\,e^{-p_{10}\\,r}\\,.\n\\end{aligned}\n\\label{eq:croadp}\\end{equation}\n\\end{linenomath*}\nParameters \\(p_i\\) of the geometry term \\(F_1\\), the moisture term \\(F_2\\), and the distance term \\(F_3\\) are given in Table~\\ref{tbl:params}. Variables \\(\\theta_\\text{field}\\) and \\(\\theta_\\text{road}\\) are given in units of \\(\\mathrm{m^3\/m^3}\\), road width \\(w\\) and distance \\(r\\) are in units of m. The function is defined for road moisture values in the range of \\(1\\leq\\theta_\\text{road}\\leq16\\,\\%\\).\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics{croad-sm.pdf}\n\\caption{\\texttt{URANOS} simulations (symbols) and correction functions \\(C_\\text{road}(\\theta_\\text{field}, \\theta_\\text{road}, w, r)\\) (lines) representing the neutron bias on roads of various widths through fields of different soil moisture. Shown for stone roads (grey) and asphalt roads (black).}\\label{fig:croadsm}\n\\end{figure}\n\n\\clearpage\n\nThe function fits well to the simulation results for different distances \\(r\\) from the road center (Fig.~\\ref{fig:croadr}), and for different \\(\\theta_\\text{field}\\), \\(\\theta_\\text{road}\\), and widths \\(w\\) (Fig.~\\ref{fig:croadsm}). However, the analytical approach shows poorer performance for road widths of \\(7\\,\\)m and beyond (not shown). The approach also overestimates the absolute bias when the field soil moisture becomes lower than the road moisture. These (rather unusual cases) should be avoided when the function is applied to roving datasets in the future. Since simulation results have indicated that the influence of slightly wetter road material is insignificant, a redefinition of the form \\(F_2(\\theta_\\text{road}>\\theta_\\text{field})=1\\) could be a sufficient approximation to these rare cases.\n\nIt is important to note that the moisture term \\(F_2\\) depends on prior knowledge of the field soil moisture \\(\\theta_\\text{field}\\).\nThe analysis of the field experiments in this work shows whether the moisture term can be replaced by a first-order approximation without prior knowledge.\n\n\\begin{figure}\n\\centering\n\\includegraphics{croad-distance.pdf}\n\\caption{\\texttt{URANOS} simulations (circles) and correction functions \\(C_\\text{road}(\\theta_\\text{field}, \\theta_\\text{road}, w, r)\\) (lines) representing the neutron bias at different distances \\(r\\) from the road center (\\(r=0\\)) for various road widths \\(w\\) (geometry shaded), field soil moisture (color), and (a) stone road material and (b) asphalt road material. Field conditions that are dryer than the road moisture (red in panel b) appear to be unresolved by the analytical approach.}\\label{fig:croadr}\n\\end{figure}\n\n\\clearpage\n\n\\begin{table}\n\\caption{Parameters \\(p_i\\) of the parameter functions \\(F_j\\) describing the road correction factor \\(C_\\text{road}\\) (eq.~\\ref{eq:croadp}), namely the geometry term \\(F_1\\), the moisture term \\(F_2\\), the distance term \\(F_3\\), and the alternative moisture term \\(F_{2'}\\) that does not require prior information about field soil moisture (eq.~\\ref{eq:croad2}).}\n\\label{tbl:params}\n\\begin{tabular}{lccccccccccc}\n\\hline\n & \\(p_0\\) & \\(p_1\\) & \\(p_2\\) & \\(p_3\\) & \\(p_4\\) & \\(p_5\\) & \\(p_6\\) & \\(p_7\\) & \\(p_8\\) & \\(p_9\\) & \\(p_{10}\\) \\\\\\hline\n\\(F_1\\) & 0.42 & 0.50 & & & & & & & & &\\\\\n\\(F_2\\) & & & 1.11 & 4.11 & 1.78 & 0.30 & & & & &\\\\\n\\(F_{2'}\\) & & & 1.06 & 4.00 & 0.16 & 0.39 & & & & &\\\\\n\\(F_3\\) & & & & & & & 0.94 & 1.10 & 2.70 & 0.06 & 0.01\\\\\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Experiment A: Estimating field soil moisture with TDR and the rover}\\label{experiment-a-estimating-field-soil-moisture-with-tdr-and-the-rover}\n\nOur first field experiment was designed in order to test the capabilities of the cosmic-ray neutron rover to capture small-scale patterns of soil moisture. During campaigns in the \\emph{Sch{\\\"a}fertal}, the rover was moved across the fields over the course of four to six hours. At the rate of one data point per minute, the technology allowed to collect more than 200--400 points in the catchment, which is an adequate number to justify ordinary kriging within the \\(1.66\\,\\mathrm{km^2}\\) area.\n\nFigure \\ref{fig:schaefertal1} shows the highly resolved CRNS soil moisture product which is able to reveal hydrological features in the catchment, such as dry hilltops, or contact springs in the valley near the creek due to shallow groundwater. Since the data were not corrected for biomass water, a probable influence of vegetation can be seen near the grove in the north-eastern part of the catchment, and possibly also near the hedgerow (south-western part). While the experiment focused on the agricultural areas of the harvested field and thus surveyed across the field and along its borders, a few roads were touched briefly at the southern and north-western hilltops, where the soil moisture appears to be slightly drier.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics{schaefertal1.pdf}\n\\caption{Soil moisture estimation by the CRNS rover in the \\emph{Sch{\\\"a}fertal} agricultural field. Data was interpolated from points (black cross) which represent the central location of the path travelled by the rover within the one minute acquisition time. Actual hydrological features like contact springs in the valley and dry hilltops are evident, but other influences of the grove, the hedgerow, and roads (see also Fig.~\\ref{fig:maps}) may distort the derived soil moisture values, indicating challenges of the method.}\\label{fig:schaefertal1}\n\\end{figure}\n\nFigure \\ref{fig:schaefertal} summarizes results from this and other field surveys in the \\emph{Sch{\\\"a}fertal} that were conducted together with a team using handheld TDR devices. Using 94 TDR samples and more than 300 rover points, it was possible to find the calibration factor \\(N_0=10447\\,\\)cph (eq.~\\ref{eq:desilets}) that explained all six sub-experiments in the catchment area.\nIn August 2015 (Fig.~\\ref{fig:schaefertal}a,b), all the fields of the \\emph{Sch{\\\"a}fertal} site were accessible with the car, however, TDR campaigns were incomplete due to technical issues. In the summer of 2014 (Fig.~\\ref{fig:schaefertal}c,d), only the northern fields could be surveyed due to agricultural activities in the southern area.\n\nFor all of the first four campaign days, Fig.~\\ref{fig:schaefertal}a--d, a good agreement between the rover and the TDR products in representing patterns and mean soil moisture in the \\emph{Sch{\\\"a}fertal} was achieved. Besides the visual impression in columns 1 and 2, the probability density functions (third column) confirm emphatically that soil moisture patterns were well captured by both methods. The two approaches appear to show remarkable agreement, despite the fact that (i) the penetration depths of both methods were different (\\(10\\,\\)cm for TDR versus 20--50\\(\\,\\)cm for CRNS), (ii) TDR data was too sparse to achieve a comparable interpolation quality, and (iii) spatially constant parameters have been used for the calibration (\\(N_0\\), \\(\\varrho_\\text{bd}\\), \\(\\theta_\\text{offset}\\)).\nIn strong contrast, rover measurements at the last two survey days, Fig.~\\ref{fig:schaefertal}e--f, show a poor agreement to field soil moisture measured by TDR. At those days, the CRNS rover had no access to the field and only crossed nearby roads and pathways. The corresponding impact on data interpretation is discussed in section~\\ref{sec:A2}.\n\nThe field campaigns highlight characteristic hydrological features, e.g.~the mentioned contact springs near the creek, that are especially prominent during dry periods and which were identified also by other researchers using conventional measurement techniques \\citep{Graeff2009, Schroeter2015}. The experiment shows that the rover can efficiently contribute to hydrological process understanding in small catchments, while the assumption of spatially constant parameters has shown to be acceptable for the almost homogeneous \\emph{Sch{\\\"a}fertal} site.\n\n\\begin{figure}\n\\centering\n\\includegraphics{schaefertal-campaigns.pdf}\n\\caption{Comparison of CRNS Rover and TDR campaigns in the \\emph{Sch{\\\"a}fertal} using interpolated data, and the probability density functions (PDF) of their overlapping area.}\\label{fig:schaefertal}\n\\end{figure}\n\n\n\\subsection{Experiment A: Taking the road effect into account}\\label{sec:A2}\n\nIn May 2014 and December 2015, the fields were cultivated and the CRNS rover surveys were restricted to the roads. Those campaigns are shown in Fig.~\\ref{fig:schaefertal}e,f, where the effect of the dry road is clearly visible in all panels.\nThis result indicates, that measurements only from the road are biased and therefore not representative for the field soil moisture. Under wet conditions, the probability density function (PDF) of soil moisture patterns becomes completely uncorrelated to the field conditions (Fig.~\\ref{fig:schaefertal}e), while under dry conditions there seems to be a simple bias of the histogram towards the dry end (Fig.~\\ref{fig:schaefertal}f).\n\nThe presented road-effect correction approach promises to account for this behavior, as it scales with the difference between road and field moisture, using information of the different types of roads crossing the catchment (Fig.~\\ref{fig:maps}). The correction function \\(C_\\text{road}\\) was applied using prior knowledge about the mean field soil moisture (eq.~\\ref{eq:croadp}). Using \\(\\theta_\\text{field}=\\langle\\theta_\\text{TDR}\\rangle\\) led to better agreement between the rover and the TDR data for both days as shown in Fig.~\\ref{fig:STcorrected} (black histograms).\n\nHowever, in most cases independent measurements of field soil moisture \\(\\theta_\\text{field}\\) are not available.\nAs an alternative, the first-order approximation of soil moisture, \\(\\theta(N)\\), using the uncorrected neutron count rate \\(N\\), could be used as a proxy to estimate the bias due to the difference of soil moisture between road and field. An alternative analytical approach for the moisture term \\(F_2\\) (eq.~\\ref{eq:croadp}) is proposed here that essentially accounts for the mismatch between \\(\\theta(N)\\) and \\(\\theta_\\text{field}\\):\n\\begin{linenomath*}\n\\begin{equation}\nF_{2'}(\\theta(N), \\theta_\\text{road})\n    \\approx p_2-p_3\\,\\theta_\\text{road}-\\frac{p_4+\\theta_\\text{road}}{p_5+\\theta(N)}\\,,\n\\label{eq:croad2}\\end{equation}\n\\end{linenomath*}\nThe updated empirical parameters \\(p_\\text{2--5}\\) (Table~\\ref{tbl:params}) have been determined based on the datasets of the \\emph{Sch{\\\"a}fertal} and another, independent experiment in the context of an interdisciplinary research project which included rover measurements across different land-use types (Scale~X, see also \\citet{Wolf2016}, data not shown here). Although the approach is empirical, the numerous tests through different sites and conditions indicate a robust potential. The corresponding probability distribution is indicated by a blue line in Fig.~\\ref{fig:STcorrected}, showing that the two approaches led to almost identical results.\n\n\\begin{figure}\n\\centering\n\\includegraphics{ST-road-corrected.pdf}\n\\caption{Application of the road correction approach on the road-only surveys in the \\emph{Sch{\\\"a}fertal} (compare Fig.~\\ref{fig:schaefertal}e--f). Patterns of the rover (left) agree well with those from TDR (middle) in terms of the probability density function (right) in the overlap area of both interpolated grids, their mean, and standard deviation. The correction is tested with two approaches of the moisture term: (1) \\(F_{2}(\\theta_\\text{field}=\\langle\\theta_\\text{TDR}\\rangle)\\) (eq.~\\ref{eq:croadp}) using the average of the TDR data (black line), and (2) \\(F_{2'}(\\theta(N))\\) (eq.~\\ref{eq:croad2}) using uncorrected neutron counts as a proxy (blue line). Kriging results using the former approach were almost identical to those using the latter approach, so that only the latter is shown in the left panel.}\\label{fig:STcorrected}\n\\end{figure}\n\n\n\\subsection{Experiment B: Road influence at a distance}\\label{experiment-b-road-influence-at-a-distance}\n\nThe experiments at the \\emph{Sheepdrove Farm} aimed to compare the soil moisture patterns of the road and the field, by surveying both compartments with the rover and excluding the one or the other during the analysis. The general objective of these experiments was to clarify whether the road correction function is able to transfer the apparent soil moisture patterns seen from the road to values that were taken in the actual field.\n\nThe road network across the farm is an ideal location to test the road effect correction, due to its wide range of road materials (gravel to asphalt) and road widths (\\(2.3\\) to \\(4.5\\,\\)m). To improve the accuracy of the rover measurements, the count rate was increased by combining multiple neutron detectors. Each of the rover datasets (experiments B and C) were compared to a stationary, well-calibrated CRNS probe and to occasional \\texttt{Theta\\ Probe} samples (not shown), in order to find a universal calibration parameter \\(N_0=11300\\,\\)cph (see also eq.~\\ref{eq:desilets}) for all datasets.\n\nIn order to rigorously test the theoretically predicted dependency of the road bias on the road moisture \\(\\theta_\\text{road}\\) and on the distance \\(r\\) to the road center, a dedicated experiment was performed at the north-west corner of the \\emph{Sheepdrove Farm} (Fig.~\\ref{fig:SFparallel}a). A gravel\/stone road (north) and an asphalt\/stone road (south) are aligned almost linearly and meet centrally at a junction. The road moisture reflects the mixture of present road materials and was estimated to be \\(\\approx11\\,\\%\\) for the asphalt\/stone mix, and \\(\\approx7\\,\\%\\) for gravel\/stone mix.\n\nThe rover measured neutrons along parallel lines in various distances from the road. For each track, the corresponding mean and standard deviation were calculated, which represent mainly the heterogeneity of soil and vegetation along each of the \\(400\\,\\)m tracks. Fig.~\\ref{fig:SFparallel}b shows how the influence of the road decreases the apparent field soil moisture as seen by the rover (left panel). Upon application of the road correction function, measurements converge to similar values for all distances (right panel) and reveal different soil moisture conditions for the nothern and southern fields. The apparent increase at \\(r=12\\,\\)m is likely caused by hydrogen present in the hedgerow and the nearby grove. The overall result provides evidence that the analytical correction function properly represents the road bias at different distances and for different materials.\n\n\\begin{figure}\n\\centering\n\\includegraphics{SF1-transect.pdf}\n\\caption{Experiment B at the \\emph{Sheepdrove Farm}. (a) Parallel tracks (dashed) at different distances from two roads of different materials that meet at the junction (\\(x=0\\), \\(y=0\\)). (b) The influence of the road decreases the apparent field soil moisture seen by the rover (left panel). Upon application of the road correction function, measurements converge to similar values for all distances (right panel) and reveal different soil moisture conditions for the nothern and southern fields. Error bars indicate the heterogeneity of water content along the whole track length of 400 m.}\\label{fig:SFparallel}\n\\end{figure}\n\n\n\\subsection{Experiment C: Patterns across roads and fields}\\label{experiment-c-patterns-across-roads-and-fields}\n\nOn three different campaign days, the roads and the surrounding fields in the central valley of the \\emph{Sheepdrove Farm} were surveyed with the CRNS rover. Road points have been corrected using eqs. \\ref{eq:croad} and \\ref{eq:croad2} based on the road types shown in Fig.~\\ref{fig:maps}. The corresponding soil moisture maps and histograms (PDFs) are shown in Fig.~\\ref{fig:sfresults}a and \\ref{fig:sfresults}b, respectively, where the three campaign days are denoted as C1, C2, and C3.\n\nIn experiment C1, it was only possible to access the borders of the field (\\(r=10\\pm5\\,\\)m) due to farming activities. Nevertheless, the correction of the road dataset led to adequate improvement of the average soil moisture distribution (Fig.~\\ref{fig:sfresults}b). However, some patterns were not adequately resolved by the road survey. According to the field measurements, the central northern field is wetter than the central southern field. From measurements on the road, only an average water content is seen with no distinction of the two fields. There are also discrepancies in the eastern part of the farm, where road and field patterns seem to be inverse. It can be speculated that one reason for this behavior is the influence of the south-eastern field, which has not been surveyed on that day. The dry spot at the north-west corner is due to buildings and a large concrete area, which were not accounted for in the correction procedure.\n\nIn experiment C2 it was possible to fully cross the fields to generate an adequate interpolation of field soil moisture. The correction of all road types appeared to agree very well with the overall pattern of the field measurements. The probability density functions show good agreement in the overlapping area of both datasets (i.e., near the road).\nThe road correction in Experiment C3 is also able to capture the patterns seen by the field survey, with the exception of the wet region in the northern part. It is speculative whether local ponds on the road, soaked soil, or local vegetation is influencing the data seen by the rover. This pathway is lowered by 1--2 meters compared to the field, and it is yet unknown whether small-scale heterogeneity of terrain features close to the sensor influences the CRNS performance. Additional vegetation correction could probably reduce the apparent soil moisture in this part, which is surrounded by unmanaged grass and hedges. In any case, low precipitation (drizzle) might have added interception water during this day, which is almost impossible to quantify.\n\n\\begin{figure}\n\\centering\n\\includegraphics{sheepdrove-results.pdf}\n\\caption{(a) Interpolated soil moisture inferred from rover measurements for experiments C1, C2, C3, showing uncorrected road data (left), corrected road data (middle), and field data (right). (b) Probability density functions of data in the overlapping areas show that the corrected road data (blue) can represent the field data (grey) better than the uncorrected road data (red). Mean and standard deviation of each distribution are provided in the top right corner.}\\label{fig:sfresults}\n\\end{figure}\n\nAll in all, the mean and standard deviation of soil moisture patterns could be restored by the application of the correction function on road points. Some of the field patterns were invisible from the road, especially when the fields on the left and on the right exhibit different water content, or when local hedges next to the roads contain or intercept water that shields the signal from the field.\n\n\\subsection{Tradeoff between measurements from the road and the field}\\label{tradeoff-between-measurements-from-the-road-and-the-field}\n\nAlthough it is evident that neutron measurements on the road can be biased substantially, it remains a challenge for experimentalists to access non-road areas, while either the access of fields is restricted or campaigns are required to cover large areas in a reasonable amount of time. Hence, roving on roads is much more practical and a necessary condition to travel from site to site. The campaigns in the \\emph{Sheepdrove Farm} combine both, road and field data, from which it could be inferred which number of measurements in the field is needed, in addition to the road data, to obtain an acceptable estimate of the field soil moisture.\n\nIn Fig.~\\ref{fig:rafaelsplot} all data points obtained during each campaign were bootstrapped, leading to more than 2000 combinations of road and field measurements.\nThe figure shows that the inclusion of uncorrected road points (red) can lead to an unreliable average value. Depending on wetness conditions, at least 80--\\(95\\,\\%\\) of the data points should be taken in the field to obtain an average that is within a two percent accuracy range around the mean field water content. In contrast to uncorrected data, corrected road data (blue) is already a good predictor for field soil moisture when any number of survey points on the road and in the field were averaged.\n\n\nThe analysis shows that any combination of field data and corrected road data can lead to a sufficient estimation of average water content in the survey area. However, the correction procedure is highly sensitive to supporting information like road moisture, field moisture, and road width (see Fig.~\\ref{fig:croadsm}). If these parameters are uncertain, their impact on the CRNS product could be substantial. The impact could be reduced by calibrating the road-correction parameters with road and field data at selected anchor locations, or by including a substantial number of field data points to the dataset measured only on roads.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics{rafael-plot3.pdf}\n\\caption{Apparent average soil moisture of 2000 combinations of road- and non-road (i.e., field) points in experiments C1--3 plotted over the fraction of selected road points in the ensemble to selected total points (i.e., sum of road and field). Data is shown for uncorrected (red) and corrected (blue) road effect. If data is not corrected, a maximum fraction of road points between 0.20 and 0.05 is acceptable (under dry or wet conditions, respectively) to provide a realistic estimate of field soil moisture within \\(2\\,\\%\\) accuracy.}\\label{fig:rafaelsplot}\n\\end{figure}\n\n\\section{Conclusions}\\label{conclusions}\n\nThe mobile cosmic-ray neutron sensor (CRNS rover) was successfully applied to estimate soil moisture at scales from a few meters to a few square kilometers. One of the most prominent insights from the detailed and extensive investigations is the confirmation that the CRNS rover is capable to capture small-scale patterns at resolutions of 10--100 m, depending on driving speed. This result opens the path for non-invasive tomography of root-zone soil moisture patterns in small catchments and agricultural fields, where traditional methods would require exhaustive and time-consuming efforts.\n\nOn the other hand, the study revealed the critical need to apply correction approaches to account for local effects of dry roads. The different experiments carried out in the course of this study showed a critical loss in the capability to estimate average field soil moisture when the field was not accessible and the measurements were taken on roads only. This effect has been quantified in this study for the first time using neutron transport simulations, and has been confirmed by dedicated experiments.\n\nWe propose an analytical correction function which accounts for various road types and soil moisture conditions. As the analytical form of the corresponding relations has been based on physical reasoning, and the parameters were determined with the help of neutron simulations, the approach could be assumed to be not site-specific and universally applicable. However, the analytical fit showed a few limitations for roads that are wetter than the field, and for road widths beyond \\(7\\,\\)m. The approach is further sensitive to the road parameters like width \\(w\\) and moisture \\(\\theta_\\text{road}\\). While the measurement of road moisture content can be impractical, this quantity could be treated as a calibration parameter by comparing data on the road and in the field at certain anchor locations.\n\nThe presented correction approach further depends on prior knowledge of field soil moisture (eqns.~\\ref{eq:croad}, \\ref{eq:croadp}). To circumvent this requirement, an adaption of the equation has been proposed that takes the uncorrected first-order approximation, \\(\\theta(N)\\), as a proxy instead (eq.~\\ref{eq:croad2}). Although we have shown its performance for the two, climatologically similar sites on ten different days throughout the years, the empirical character of this alternative approach requires more test cases under more various sites and conditions.\n\nThe corrected road data has been compared with field soil moisture inferred from independent TDR (experiment A) and rover measurements (experiments B and C). In all cases the corrected soil moisture product sensed from the road showed remarkable agreement with the patterns, the mean, and the standard deviation of soil moisture in the field. However, a few limitations have been identified. If strong differences in soil moisture are present between neighboring fields passed by the rover, it may be not possible for the sensor to capture the corresponding patterns due to the isotropic nature of neutron detection. Moreover, local ponds on pathways or nearby unmanaged vegetation could further influence the neutron signal in a way that is not representative for the field behind.\n\nNevertheless, a considerable amount of uncertainty is introduced to measurements from roads due to the high contribution of non-field neutrons and the uncertain properties of the road and its surroundings. Therefore, it is advisable to drive directly on the field wherever possible, or to take additional measurements on the field every now and then. This is advisable not only to make sure that the parameters of the road correction lead to a proper representation of field soil moisture, but also to support spatial interpolation.\n\nBased on the conclusions above, we generally recommend to correct for the road effect before spatial CRNS data is used to support hydrological models or agricultural decisions. With regards to evaluation of remote-sensing products \\citep[e.g.,][]{Chrisman2013} dry roads are also part of the remotely sensed average soil moisture, so that different correction approaches might be needed to compare both area-averaged products. There might also be ways to reduce the contribution of the roads in future developments of the neutron detector. For example, by mounting the detector on top of the car where it is more exposed to far-field neutrons.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\acknowledgments\nData is available from the authors on request. MS, RR, and JI thank Dan Bull for providing access to the Sheepdrove Organic Farm. MS, IS, and UW thank Thomas Grau, Mandy Kasner, and Andreas Schmidt for their support during field campaigns in the Sch{\\\"a}fertal. MS acknowledges kind support by the Helmholtz Impulse and Networking Fund through Helmholtz Interdisciplinary School for Environmental Research (HIGRADE). JI is funded by the Queen's School of Engineering, University of Bristol, EPSRC, grant code: EP\/L504919\/1. RR, JI and Sheepdrove Organic Farm activities are funded by the Natural Environment Research Council (A MUlti-scale Soil moistureEvapotranspiration Dynamics study (AMUSED); grant number NE\/M003086\/1). The research was funded and supported by the Terrestrial Environmental Observatories (TERENO), which is a joint collaboration program involving several Helmholtz Research Centers in Germany.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}