bluuebunny/crossref_metadata_2025 huggingface dataset:
Real Schottky Uniformizations and Jacobians of May Surfaces
Authors: Hidalgo Rubén A. , Rodríguez Rubí E.
Published: December 2011

Given a closed Riemann surface

R

of genus

p \geq 2

together with an anticonformal involution

\tau:R \to R

with fixed points, we consider the group

K(R,\tau)

consisting of the conformal and anticonformal automorphisms of

R

which commute with

\tau

. It is a well known fact due to C. L. May that the order of

K(R,\tau)

is at most

24(p-1)

and that such an upper bound is attained for infinitely many, but not all, values of

p

. May also proved that for every genus

p \geq 2

there are surfaces for which the order of

K(R,\tau)

can be chosen to be

8p

and

8(p+1)

. These type of surfaces are called \textit{May surfaces}. In this note we construct real Schottky uniformizations of every May surface. In particular, the corresponding group

K(R,\tau)

lifts to such an uniformization. With the help of these real Schottky uniformizations, we obtain (extended) symplectic representations of the groups

K(R,\tau)

. We study the families of principally polarized abelian varieties admitting the given group of automorphisms and compute the corresponding Riemann matrices, including those for the Jacobians of May surfaces.
An analysis of quantum Fokker–Planck models: A Wigner function approach
Authors: Arnold Anton , López José L. , Markowich Peter A. , Soler Juan
Published: December 2011

The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator.
Clifford and Harmonic Analysis on Cylinders and Tori
Authors: Kraußhar Rolf Sören , Ryan John
Published: December 2011

Cotangent type functions in

\mathbb{R}^n

are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds

\mathbb{R}^n / \mathbb{Z}^k

where

1\leq k\leq n

. Basic properties of these kernels are discussed including introducing a Cauchy formula, Green's formula, Cauchy transform, Poisson kernel, Szegö kernel and Bergman kernel for certain types of domains. Singular Cauchy integrals are also introduced as are associated Plemelj projection operators. These in turn are used to study Hardy spaces in this context. Also the analogues of Calderón-Zygmund type operators are introduced in this context, together with singular Clifford holomorphic, or monogenic, kernels defined on sector domains in the context of cylinders. Fundamental differences in the context of the

n

-torus arising from a double singularity for the generalized Cauchy kernel on the torus are also discussed.
Quasinormal Families of Meromorphic Functions
Authors: Pang Xuecheng , Nevo Shahar , Zalcman Lawrence
Published: December 2011

Let

\mathcal{F}

be a family of functions meromorphic on the plane domain

D

, all of whose zeros are multiple. Suppose that

f'(z)\ne 1

for all

f\in \mathcal{F}

and

z\in D.

Then if

\mathcal{F}

is quasinormal on

D

, it is quasinormal of order 1 there.
Interpolation and extrapolation of smooth functions by linear operators
Authors: Fefferman Charles
Published: December 2011

Let

C^{m , 1} ( \mathbb{R}^n)

be the space of functions on

\mathbb{R}^n

whose

m^{\sf th}

derivatives are Lipschitz 1. For

E \subset \mathbb{R}^n

, let

C^{m , 1} (E)

be the space of all restrictions to

E

of functions in

C^{m,1} ( \mathbb{R}^n)

. We show that there exists a bounded linear operator

T: C^{m , 1} (E) \rightarrow C^{m , 1} ( \mathbb{R}^n)

such that, for any

f \in C^{m , 1} ( E )

, we have

T f = f

on

E

.
A Simplified Proof of Desingularization and Applications
Authors: Bravo Ana María , Encinas Santiago , Villamayor U. Orlando
Published: December 2011

This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i.e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme

X

is achieved by using the procedure of principalization for the ideal

{\mathcal I}(X)

associated to the embedded scheme

X

. The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof.
Asymptotic windings over the trefoil knot
Authors: Franchi Jacques
Published: December 2011

Consider the group

G:=PSL_2(\mathbb R)

and its subgroups

\Gamma:= PSL_2(\mathbb{Z})

and

\Gamma':= DSL_2(\mathbb{Z})

.

G/\Gamma

is a canonical realization (up to an homeomorphism) of the complement

\mathbb S^3\setminus T

of the trefoil knot

T

, and

G/\Gamma'

is a canonical realization of the 6-fold branched cyclic cover of

\mathbb S^3\setminus T

, which has 3-dimensional cohomology of 1-forms. Putting natural left-invariant Riemannian metrics on

G

, it makes sense to ask which is the asymptotic homology performed by the Brownian motion in

G/\Gamma'

, describing thereby in an intrinsic way part of the asymptotic Brownian behavior in the fundamental group of the complement of the trefoil knot. A good basis of the cohomology of

G/\Gamma'

, made of harmonic 1-forms, is calculated, and then the asymptotic Brownian behavior is obtained, by means of the joint asymptotic law of the integrals of the above basis along the Brownian paths. Finally the geodesics of

G

are determined, a natural class of ergodic measures for the geodesic flow is exhibited, and the asymptotic geodesic behavior in

G/\Gamma'

is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).
Solution to the gradient problem of C. E. Weil
Authors: Buczolich Zoltán
Published: December 2011

In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set

G\subset \mathbb{R}^{2}

we construct a differentiable function

f:G\to\mathbb{R}

for which there exists an open set

\Omega_{1}\subset\mathbb{R}^{2}

such that

\nabla f(\mathbf{p})\in \Omega_{1}

for a

\mathbf{p}\in G

but

\nabla f(\mathbf{q})\not\in\Omega_{1}

for almost every

\mathbf{q}\in G

. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.
High order regularity for subelliptic operators on Lie groups of polynomial growth
Authors: Dungey Nick
Published: December 2011

Let

G

be a Lie group of polynomial volume growth, with Lie algebra

{\mathfrak g}

. Consider a second-order, right-invariant, subelliptic differential operator

H

on

G

, and the associated semigroup

S_t = e^{-tH}

. We identify an ideal

{\mathfrak n}'

of

{\mathfrak g}

such that

H

satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of

{\mathfrak n}'

. The regularity is expressed as

L_2

estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel. We obtain the boundedness in

L_p

,

1<p<\infty

, of some associated Riesz transform operators. Finally, we show that

{\mathfrak n}'

is the largest ideal of

{\mathfrak g}

for which the regularity results hold. Various algebraic characterizations of

{\mathfrak n}'

are given. In particular,

{\mathfrak n}'= {\mathfrak s}\oplus {\mathfrak n}

where

{\mathfrak n}

is the nilradical of

{\mathfrak g}

and

{\mathfrak s}

is the largest semisimple ideal of

{\mathfrak g}

. Additional features of this article include an exposition of the structure theory for

G

in Section 2, and a concept of twisted multiplications on Lie groups which includes semidirect products in the Appendix.
Approximation in law to the $d$-parameter fractional Brownian sheet based on the functional invariance principle
Authors: Bardina Xavier , Florit Carme
Published: December 2011

We show a result of approximation in law of the

d

-parameter fractional Brownian sheet in the space of the continuous functions on

[0,T]^d

. The construction of these approximations is based on the functional invariance principle.
$L^p$ decay estimates for weighted oscillatory integral operators on $\mathbb{R}$
Authors: Pramanik Malabika , Yang Chan Woo
Published: December 2011

In this paper, we formulate necessary conditions for decay rates of

L^p

operator norms of weighted oscillatory integral operators on

\mathbb{R}

and give sharp

L^2

estimates and nearly sharp

L^p

estimates.
Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one
Authors: Sobolev Alexander V.
Published: December 2011

We consider a periodic pseudo-differential operator on the real line, which is a lower-order perturbation of an elliptic operator with a homogeneous symbol and constant coefficients. It is proved that the density of states of such an operator admits a complete asymptotic expansion at large energies. A few first terms of this expansion are found in a closed form.
Compact embeddings of Brézis-Wainger type
Authors: Cobos Fernando , Kühn Thomas , Schonbek Tomas
Published: December 2011

Let

\Omega

be a bounded domain in

\mathbb R^n

and denote by

id_\Omega

the restriction operator from the Besov space

B_{pq}^{1+n/p}(\mathbb R^n)

into the generalized Lipschitz space

Lip^{(1,-\alpha)}(\Omega)

. We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like

e_k(id_\Omega) \sim k^{-1/p}

if

\alpha > \max (1+2/p-1/q,1/p)

. Our estimates improve previous results by Edmunds and Haroske.
Genus 3 normal coverings of the Riemann sphere branched over 4 points
Authors: Fuertes Yolanda , Streit Manfred
Published: December 2011

In this paper we study the 5 families of genus 3 compact Riemann surfaces which are normal coverings of the Riemann sphere branched over 4 points from very different aspects: their moduli spaces, the uniform Belyi functions that factorize through the quotient by the automorphism groups and the Weierstrass points of the non hyperelliptic families.
A geometry on the space of probabilities I. The finite dimensional case
Authors: Gzyl Henryk , Recht Lázaro
Published: December 2011

In this note we provide a natural way of defining exponential coordinates on the class of probabilities on the set

\Omega = [1,n]

or on

\mathbb{P} = \{p=(p_1,\dots,p_n)\in \mathbb{R}^n | p_i > 0; \Sigma_{i=1}^n p_i =1\}

. For that we have to regard

\mathbb{P}

as a projective space and the exponential coordinates will be related to geodesic flows in

\mathbb{C}^n

.
How smooth is almost every function in a Sobolev space?
Authors: Fraysse Aurélia , Jaffard Stéphane
Published: December 2011

We show that almost every function (in the sense of prevalence) in a Sobolev space is multifractal: Its regularity changes from point to point; the sets of points with a given Hölder regularity are fractal sets, and we determine their Hausdorff dimension.
Le calcul fonctionnel sous-linéaire dans les espaces de Besov homogènes
Authors: Bourdaud Gérard , Meyer Yves
Published: December 2011

On établit l'estimation sous-linéaire

\|f \circ g\| \leq c(f) \|g\|

, la norme étant celle de l'espace de Besov homogène

\dot B^{s,q}_{p}(\mathbb{R}^n)

, où

1\leq p<+\infty

,

1\leq q\leq +\infty

et

0<s<1+(1/p)

. La fonction

f

est supposée appartenir à une classe

U^1_p

introduite antérieurement par Bourdaud et Kateb, qui contient notamment les primitives des fonctions à variation bornée sur

{\mathbb R}

. À titre d'application, on montre qu'une fonction étagée appartient à

\dot{B}^{1,\infty}_1({\mathbb R^2})

si et seulement si elle appartient à

BV({\mathbb R^2})

.
Some non-linear function theoretic properties of Riemannian manifolds
Authors: Pigola Stefano , Rigoli Marco , Setti Alberto G.
Published: December 2011

We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the

p

-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators.
Asymptotic behaviour of monomial ideals on regular sequences
Authors: Sedghi Monireh
Published: December 2011

Let

R

be a commutative Noetherian ring, and let

\mathbf{x}= x_1, \ldots, x_d

be a regular

R

-sequence contained in the Jacobson radical of

R

. An ideal

I

of

R

is said to be a monomial ideal with respect to

\mathbf{x}

if it is generated by a set of monomials

x_1^{e_1}\ldots x_d^{e_d}

. The monomial closure of

I

, denoted by

\widetilde{I}

, is defined to be the ideal generated by the set of all monomials

m

such that

m^n\in I^n

for some

n\in \mathbb{N}

. It is shown that the sequences

\mathrm{Ass}_RR/\widetilde{I^n}

and

\mathrm{Ass}_R\widetilde{I^n}/I^n

,

n=1,2, \ldots,

of associated prime ideals are increasing and ultimately constant for large

n

. In addition, some results about the monomial ideals and their integral closures are included.
Restricted Radon Transforms and Unions of Hyperplanes
Authors: Oberlin Daniel M.
Published: December 2011

We study

L^p (\mathbb{R}^n)\rightarrow L^{\alpha ,\infty}_{d\mu (\sigma)}(L^{\infty}_{dt})

estimates for the Radon transform in certain cases where the dimension of the measure

\mu

on

\Sigma^{(n-1)}

is less than

n-1

.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
Authors: Barthe Franck , Cattiaux Patrick , Roberto Cyril
Published: December 2011

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general

F

-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the integrability improving properties of the Heat semigroup associated to the Boltzmann measures

\mu_{\alpha}(dx) = (Z_{\alpha})^{-1} e^{-2|x|^{\alpha}} dx

, when

\alpha\in (1,2)

. As an application we derive accurate isoperimetric inequalities for their products. This completes earlier works by Bobkov-Houdré and Talagrand, and provides a scale of dimension free isoperimetric inequalities as well as comparison theorems.
Poisson kernels of half–spaces in real hyperbolic spaces
Authors: Byczkowski Tomasz , Graczyk Piotr , Stós Andrzej
Published: December 2011

We provide an integral formula for the Poisson kernel of half-spaces for Brownian motion in real hyperbolic space

\mathbb{H}^n

. This enables us to find asymptotic properties of the kernel. We also show convergence to the Poisson kernel of the whole space

\mathbb{H}^n

. For

n=3

,

4

or

6

we compute explicit formulas for the Poisson kernel itself.
On minimal non-supersoluble groups
Authors: Ballester-Bolinches Adolfo , Esteban-Romero Ramón
Published: December 2011

The aim of this paper is to classify the finite minimal non-

p

-supersoluble groups,

p

a prime number, in the

p

-soluble universe.
Convex functions on Carnot groups
Authors: Juutinen Petri , Lu Guozhen , Manfredi Juan J. , Stroffolini Bianca
Published: December 2011

We consider the definition and regularity properties of convex functions in Carnot groups. We show that various notions of convexity in the subelliptic setting that have appeared in the literature are equivalent. Our point of view is based on thinking of convex functions as subsolutions of homogeneous elliptic equations.
Small gaps in coefficients of $L$-functions and $\mathfrak{B}$-free numbers in short intervals
Authors: Kowalski Emmanuel , Robert Olivier , Wu Jie
Published: December 2011

We discuss questions related to the non-existence of gaps in the series defining modular forms and other arithmetic functions of various types, and improve results of Serre, Balog and Ono, and Alkan using new results about exponential sums and the distribution of

\mathfrak{B}

-free numbers.
On a Parabolic Symmetry Problem
Authors: Lewis John L. , Nyström Kaj
Published: December 2011

In this paper we prove a symmetry theorem for the Green function associated to the heat equation in a certain class of bounded domains

\Omega\subset\mathbb{R}^{n+1}

. For

T>0

, let

\Omega_T=\Omega\cap[\mathbb{R}^n\times (0,T)]

and let

G

be the Green function of

\Omega_T

with pole at

(0,0)\in\partial_p\Omega_T

. Assume that the adjoint caloric measure in

\Omega_T

defined with respect to

(0,0)

,

\hat\omega

, is absolutely continuous with respect to a certain surface measure,

\sigma

, on

\partial_p\Omega_T

. Our main result states that if

\frac {d\hat\omega}{d\sigma}(X,t)=\lambda\frac {|X|}{2t}

for all

(X,t)\in \partial_p\Omega_T\setminus\{(X,t): t=0\}

and for some

\lambda>0

, then

\partial_p\Omega_T\subseteq\{(X,t):W(X,t)=\lambda\}

where

W(X,t)

is the heat kernel and

G=W-\lambda

in

\Omega_T

. This result has previously been proven by Lewis and Vogel under stronger assumptions on

\Omega

.
$C^1$ extensions of functions and stabilization of Glaeser refinements
Authors: Klartag Bo'az , Zobin Nahum
Published: December 2011

Given an arbitrary set

E \subset \mathbb R^n

,

n\ge 2

, and a function

f: E \rightarrow \mathbb R

, consider the problem of extending

f

to a

C^1

function defined on the entire

\mathbb R^n

. A procedure for determining whether such an extension exists was suggested in 1958 by G. Glaeser. In 2004 C. Fefferman proposed a related procedure for dealing with the much more difficult cases of higher smoothness. The procedures in question require iterated computations of some bundles until the bundles stabilize. How many iterations are needed? We give a sharp estimate for the number of iterations that could be required in the

C^1

case. Some related questions are discussed.
Percolation, Perimetry, Planarity
Authors: Kozma Gady
Published: December 2011

Let

G

be a planar graph with polynomial growth and isoperimetric dimension bigger than

1

. Then the critical

p

for Bernoulli percolation on

G

satisfies

p_{c} < 1

.
Quasi-similarity of contractions having a 2 × 1 characteristic function
Authors: Bermudo Sergio , Mancera Carmen H. , Paúl Pedro J. , Vasyunin Vasily
Published: December 2011

Let

T_1 \in \mathscr B( \mathscr H_1)

be a completely non-unitary contraction having a non-zero characteristic function

\Theta_1

which is a

2 \times 1

column vector of functions in

H^\infty

. As it is well-known, such a function

\Theta_1

can be written as

\Theta_1=w_1 m_1 \left[ {a_1} \atop {b_1} \right]

where

w_1, m_1, a_1, b_1 \in H^\infty

are such that

w_1

is an outer function with

|w_1|\leq 1

,

m_1

is an inner function,

|a_1|^2 + |b_1|^2 =1

, and

a_1 \wedge b_1 = 1

(here

\wedge

stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction

T_2 \in \mathscr B( \mathscr H_2)

having also a

2 \times 1

characteristic function

\Theta_2=w_2 m_2 \left[ {a_2} \atop {b_2} \right]

. We prove that

T_1

is quasi-similar to

T_2

if, and only if, the following conditions hold:
Measurability of equivalence classes and MEC$_p$-property in metric spaces
Authors: Järvenpää Esa , Järvenpää Maarit , Rogovin Kevin , Rogovin Sari , Shanmugalingam Nageswari
Published: December 2011

We prove that a locally compact metric space that supports a doubling measure and a weak

p

-Poincaré inequality for some

1\le p < \infty

is a

\mathrm{MEC}_p

-space. The methods developed for this purpose include measurability considerations and lead to interesting consequences. For example, we verify that each extended real valued function having a

p

-integrable upper gradient is locally

p

-integrable.
A new type of solutions for a singularly perturbed elliptic Neumann problem
Authors: Li Gongbao , Peng Shuangjie , Yan Shusen
Published: December 2011

We prove the existence of positive solutions concentrating simultaneously on some higher dimensional manifolds near and on the boundary of the domain for a nonlinear singularly perturbed elliptic Neumann problem.
Heat kernel transform for nilmanifolds associated to the Heisenberg group
Authors: Krötz Bernhard , Thangavelu Sundaram , Xu Yuan
Published: December 2011

We study the heat kernel transform on a nilmanifold

M

of the Heisenberg group. We show that the image of

L^2(M)

under this transform is a direct sum of weighted Bergman spaces which are related to twisted Bergman and Hermite-Bergman spaces.
On the verbal width of finitely generated pro-$p$ groups
Authors: Jaikin-Zapirain Andrei
Published: December 2011

Let

p

be a prime. It is proved that a non-trivial word

w

from a free group

F

has finite width in every finitely generated pro-

p

group if and only if

w\not \in (F^\prime)^{p} F^{\prime\prime}

. Also it is shown that any word

w

has finite width in a compact

p

-adic group.
Projections of hypersurfaces in the hyperbolic space to hyperhorospheres and hyperplanes
Authors: Izumiya Shyuichi , Tari Farid
Published: December 2011

We study in this paper orthogonal projections in a hyperbolic space to hyperhorospheres and hyperplanes. We deal in more details with the case of embedded surfaces

M

in

H^3_+(-1)

. We study the generic singularities of the projections of

M

to horospheres and planes. We give geometric characterizations of these singularities and prove duality results concerning the bifurcation sets of the families of projections. We also prove Koenderink type theorems that give the curvature of the surface in terms of the curvatures of the profile and the normal section of the surface.
Homology exponents for $H$-spaces
Authors: Clément Alain , Scherer Jérôme
Published: December 2011

We say that a space

X

admits a \emph{homology exponent} if there exists an exponent for the torsion subgroup of

H^*(X;\mathbb Z)

. Our main result states that if an

H

-space of finite type admits a homology exponent, then either it is, up to

2

-completion, a product of spaces of the form

B\mathbb Z/2^r

,

S^1

,

\mathbb C P^\infty

, and

K(\mathbb Z,3)

, or it has infinitely many non-trivial homotopy groups and

k

-invariants. Relying on recent advances in the theory of

H

-spaces, we then show that simply connected

H

-spaces whose mod

2

cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod

2

finite

H

-spaces with copies of

\mathbb C P^\infty

and

K(\mathbb Z,3)

.
Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term
Authors: Polidoro Sergio , Ragusa Maria Alessandra
Published: December 2011

We prove a Harnack inequality for the positive solutions of ultraparabolic equations of the type

\mathcal {L}_0 u + \mathcal {V} u = 0,

where

\mathcal {L}_0

is a linear second order hypoelliptic operator and

\mathcal {V}

belongs to a class of functions of Stummel-Kato type. We also obtain the existence of a Green function and an uniqueness result for the Cauchy-Dirichlet problem.
Majorizing measures and proportional subsets of bounded orthonormal systems
Authors: Guédon Olivier , Mendelson Shahar , Pajor Alain , Tomczak-Jaegermann Nicole
Published: December 2011

In this article we prove that for any orthonormal system

(\varphi_j)_{j=1}^n \subset L_2

that is bounded in

L_{\infty}

, and any

1 < k < n

, there exists a subset

I

of cardinality greater than

n-k

such that on

\mathrm{span}\{\varphi_i\}_{i \in I}

, the

L_1

norm and the

L_2

norm are equivalent up to a factor

\mu (\log \mu)^{5/2}

, where

\mu = \sqrt{n/k} \sqrt{\log k}

. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.
Bi-Lipschitz decomposition of Lipschitz functions into a metric space
Authors: Schul Raanan
Published: December 2011

We prove a quantitative version of the following statement. Given a Lipschitz function

f

from the k-dimensional unit cube into a general metric space, one can be decomposed

f

into a finite number of BiLipschitz functions

f|_{F_i}

so that the k-Hausdorff content of

f([0,1]^k\smallsetminus \cup F_i)

is small. We thus generalize a theorem of P. Jones [Lipschitz and bi-Lipschitz functions. Rev. Mat. Iberoamericana 4 (1988), no. 1, 115–121] from the setting of

\mathbb{R}^d

to the setting of a general metric space. This positively answers problem 11.13 in “Fractured Fractals and Broken Dreams” by G. David and S. Semmes, or equivalently, question 9 from “Thirty-three yes or no questions about mappings, measures, and metrics” by J. Heinonen and S. Semmes. Our statements extend to the case of
coarse
Lipschitz functions.
Adams inequality on metric measure spaces
Authors: Mäkäläinen Tero
Published: December 2011

In this paper, we prove the Adams inequality in complete metric spaces supporting a Poincaré inequality with a doubling measure. We also prove the trace inequalities for the Riesz potentials.
The sharp $A_p$ constant for weights in a reverse-Hölder class
Authors: Dindoš Martin , Wall Treven
Published: December 2011

Coifman and Fefferman established that the class of Muckenhoupt weights is equivalent to the class of weights satisfying the "reverse Hölder inequality". In a recent paper V. Vasyunin [The exact constant in the inverse Hölder inequality for Muckenhoupt weights. St. Petersburg Math. J. 15 (2004), no. 1, 49-79] presented a proof of the reverse Hölder inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function technique to find the sharp

A_p

constants for weights in a reverse-Hölder class on an interval; we also find the sharp constants for the higher-integrability result of Gehring [The

L_p

-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265-277]. Additionally, we find sharp bounds for the

A_p

constants of reverse-Hölder-class weights defined on rectangles in

\mathbb{R}^n

, as well as bounds on the

A_p

constants for reverse-Hölder weights defined on cubes in

\mathbb{R}^n

, without claiming the sharpness.
$h^1$, bmo, blo and Littlewood-Paley $g$-functions with non-doubling measures
Authors: Hu Guoen , Yang Dachun , Yang Dongyong
Published: December 2011

Let

\mu

be a nonnegative Radon measure on

{\mathbb R}^d

which satisfies the growth condition that there exist constants

C_0 > 0

and

n\in(0,d]

such that for all

x\in{\mathbb R}^d

and

r > 0

,

\mu(B(x,\,r)) \le C_0 r^n

, where

B(x,r)

is the open ball centered at

x

and having radius

r

. In this paper, we introduce a local atomic Hardy space

{h_{\rm atb}^{1,\infty}(\mu)}

, a local BMO-type space

{\mathop\mathrm{rbmo}(\mu)}

and a local BLO-type space

{\mathop\mathrm{rblo}(\mu)}

in the spirit of Goldberg and establish some useful characterizations for these spaces. Especially, we prove that the space

{\mathop\mathrm{rbmo}(\mu)}

satisfies a John-Nirenberg inequality and its predual is

{h_{\rm atb}^{1,\infty}(\mu)}

. We also establish some useful properties of

{\mathop\mathrm{RBLO}\,(\mu)}

and improve the known characterization theorems of

{\mathop\mathrm{RBLO}(\mu)}

in terms of the natural maximal function by removing the assumption on the regularity condition. Moreover, the relations of these local spaces with known corresponding function spaces are also presented. As applications, we prove that the inhomogeneous Littlewood-Paley

g

-function

g(f)

of Tolsa is bounded from

{h_{\rm atb}^{1,\infty}(\mu)}

to

{L^1(\mu)}

, and that

[g(f)]^2

is bounded from

{\mathop\mathrm{rbmo}(\mu)}

to

{\mathop\mathrm{rblo}(\mu)}

.
The Nullstellensatz for real coherent analytic surfaces
Authors: Broglia Fabrizio , Pieroni Federica
Published: December 2011

In this paper we prove Hilbert Nullstellensatz for real coherent analytic surfaces and we give a precise description of the obstruction to get it in general. Refering the first, we prove that the ideals of global functions vanishing on analytic subsets are exactly the real saturated ones. For

\mathbb{R}^3

we prove that the real Nullstellensatz holds for real saturated ideals if and only if no principal ideal generated by a function whose zero set is a curve (indeed, a special function) is real. This led us to compare the Nullstellensatz problem with the Hilbert 17th one, also in its weaker form involving infinite sums of squares, proving that they share in fact the same obstruction.
Haar multipliers meet Bellman functions
Authors: Pereyra María Cristina
Published: December 2011

Using Bellman function techniques, we obtain the optimal dependence of the operator norms in

L^2(\mathbb{R})

of the Haar multipliers

T_w^t

on the corresponding

RH^d_2

or

A^d_2

characteristic of the weight

w

, for

t=1,\pm 1/2

. These results can be viewed as particular cases of estimates on homogeneous spaces

L^2(vd\sigma)

, for

\sigma

a doubling positive measure and

v\in A^d_2(d\sigma)

, of the weighted dyadic square function

S_{\sigma}^d

. We show that the operator norms of such square functions in

L^2(v d\sigma)

are bounded by a linear function of the

A^d_2(d\sigma )

characteristic of the weight

v

, where the constant depends only on the doubling constant of the measure

\sigma

. We also show an inverse estimate for

S_{\sigma}^d

. Both results are known when

d\sigma=dx

. We deduce both estimates from an estimate for the Haar multiplier

(T_v^{\sigma})^{1/2}

on

L^2(d\sigma)

when

v\in A_2^d(d\sigma)

, which mirrors the estimate for

T_w^{1/2}

in

L^2(\mathbb{R})

when

w\in A^d_2

. The estimate for the Haar multiplier adapted to the

\sigma

measure,

(T_v^{\sigma})^{1/2}

, is proved using Bellman functions. These estimates are sharp in the sense that the rates cannot be improved and be expected to hold for all

\sigma

, since the particular case

d\sigma=dx

,

v=w

, correspond to the estimates for the Haar multipliers

T^{1/2}_w

proven to be sharp.
Threshold solutions for the focusing 3D cubic Schrödinger equation
Authors: Duyckaerts Thomas , Roudenko Svetlana
Published: December 2011

We study the focusing 3d cubic NLS equation with

H^1

data at the mass-energy threshold, namely, when

M[u_0]E[u_0]{=}M[Q]E[Q]

. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) was classified when

M[u_0]E[u_0] < M[Q]E[Q]

. In this paper, we first exhibit 3 special solutions:

e^{it} Q

and

Q^\pm

, where

Q

is the ground state,

Q^\pm

exponentially approach the ground state solution in the positive time direction,

Q^+

has finite time blow up and

Q^-

scatters in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to

\dot{H}^{1/2}

symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schrödinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.
Vector-valued distributions and Hardy’s uncertainty principle for operators
Authors: Cowling Michael , Demange Bruno , Sundari M.
Published: December 2011

Suppose that

f

is a function on

\mathbb{R}^n

such that

\exp(a |\cdot|^2) f

and

\exp(b |\cdot|^2) \hat f

are bounded, where

a,b > 0

. Hardy's Uncertainty Principle asserts that if

ab > \pi^2

, then

f = 0

, while if

ab = \pi^2

, then

f = c\exp(-a|\cdot|^2)

. In this paper, we generalise this uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.
Convergence of metric graphs and energy forms
Authors: Kasue Atsushi
Published: December 2011

In this paper, we begin with clarifying spaces obtained as limits of sequences of finite networks from an analytic point of view, and we discuss convergence of finite networks with respect to the topology of both the Gromov-Hausdorff distance and variational convergence called

\Gamma

-convergence. Relevantly to convergence of finite networks to infinite ones, we investigate the space of harmonic functions of finite Dirichlet sums on infinite networks and their Kuramochi compactifications.
Riesz transforms on forms and $L^p$-Hodge decomposition on complete Riemannian manifolds
Authors: Li Xiang-Dong
Published: December 2011

In this paper we prove the Strong

L^p

-stability of the heat semigroup generated by the Hodge Laplacian on complete Riemannian manifolds with non-negative Weitzenböck curvature. Based on a probabilistic representation formula, we obtain an explicit upper bound of the

L^p

-norm of the Riesz transforms on forms on complete Riemannian manifolds with suitable curvature conditions. Moreover, we establish the Weak

L^p

-Hodge decomposition theorem on complete Riemannian manifolds with non-negative Weitzenböck curvature.
A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps
Authors: Chen Zhen-Qing , Kumagai Takashi
Published: December 2011

In this paper, we consider the following type of non-local (pseudo-differential) operators

\mathcal{L}

on

\mathbb{R}^d

:

\begin{align*}
\mathcal{L} u(x) =& \frac{1}{2} \sum_{i, j=1}^d \frac{\partial}{\partial x_i} \Big(a_{ij}(x) \frac{\partial u(x)}{\partial x_j}\Big) \\
&\ + \lim_{\varepsilon \downarrow 0} \int_{\{y\in \mathbb{R}^d: |y-x|>\varepsilon\}} (u(y)-u(x)) J(x, y) dy,
\end{align*}

where

A(x)=(a_{ij}(x))_{1\leq i,j\leq d}

is a measurable

d\times d

matrix-valued function on

\mathbb{R}^d

that is uniformly elliptic and bounded and

J

is a symmetric measurable non-trivial non-negative kernel on

\mathbb{R}^d \times \mathbb{R}^d

satisfying certain conditions. Corresponding to

\mathcal{L}

is a symmetric strong Markov process

X

on

\mathbb{R}^d

that has both the diffusion component and pure jump component. We establish a priori Hölder estimate for bounded parabolic functions of

\mathcal{L}

and parabolic Harnack principle for positive parabolic functions of

\mathcal{L}

. Moreover, two-sided sharp heat kernel estimates are derived for such operator

\mathcal{L}

and jump-diffusion

X

. In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on

\mathbb{R}^d

. To establish these results, we employ methods from both probability theory and analysis.
Socle theory for Leavitt path algebras of arbitrary graphs
Authors: Aranda Pino Gonzalo , Martín Barquero Dolores , Martín González Cándido , Siles Molina Mercedes
Published: December 2011

The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate the socle of a Leavitt path algebra. A concrete description of the socle of a Leavitt path algebra is obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field. New proofs of the Graded Uniqueness and of the Cuntz-Krieger Uniqueness Theorems are given, by using very different means.
Bernstein-Heinz-Chern results in calibrated manifolds
Authors: Li Guanghan , Salavessa Isabel M.C.
Published: December 2011

Given a calibrated Riemannian manifold

\overline{M}

with parallel calibration

\Omega

of rank

m

and

M

an orientable m-submanifold with parallel mean curvature

H

, we prove that if

\cos\theta

is bounded away from zero, where

\theta

is the

\Omega

-angle of

M

, and if

M

has zero Cheeger constant, then

M

is minimal. In the particular case

M

is complete with

Ricci^M\geq 0

we may replace the boundedness condition on

\cos\theta

by

\cos\theta\geq Cr^{-\beta}

, when

r\rightarrow+\infty

, where

0 < \beta < 1

and

C > 0

are constants and

r

is the distance function to a point in

M

. Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on an estimation of

\|H\|

in terms of

\cos\theta

and an isoperimetric inequality. In a similar way, we also give some conditions to conclude

M

is totally geodesic. We study some particular cases.
Le Théorème du symbole total d'un opérateur différentiel $p$-adique
Authors: Mebkhout Zoghman , Narváez Macarro Luis
Published: December 2011

Let

{\mathcal X}^\dagger

be a smooth

\dagger

-scheme (in the sense of Meredith) over a complete discrete valuation ring

(V, {\mathfrak m})

of unequal characteristics

(0,p)

and let

{\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}

be the sheaf of

V

-linear endomorphisms of

{\mathcal O}_{{\mathcal X}^\dagger}

whose reduction modulo

{\mathfrak m}^s

is a linear differential operator of order bounded by an affine function in

s

. In this paper we prove that locally there is an

{\mathcal O}_{{\mathcal X}^\dagger}

-isomorphism between the sections of

{\mathcal D}^\dagger_{{\mathcal X}^\dagger/V}

and the overconvergent total symbols, and we deduce a cohomological triviality property.
Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities
Authors: Chen Qionglei , Miao Changxing , Zhang Zhifei
Published: December 2011

In this paper, we prove the local well-posedness in critical Besov spaces for the compressible Navier-Stokes equations with density dependent viscosities under the assumption that the initial density is bounded away from zero.
On some maximal multipliers in $L^p$
Authors: Demeter Ciprian
Published: December 2011

We extend an

L^2

maximal multiplier result of Bourgain to all

L^p

spaces,

1 < p < \infty

.
Overdetermined problems in unbounded domains with Lipschitz singularities
Authors: Farina Alberto , Valdinoci Enrico
Published: December 2011

We study the overdetermined problem

\left\{ \begin{array}{ll} \Delta u + f(u) = 0 & \text{ in $\Omega$,} \\ u = 0 & \text{ on $\partial\Omega$,} \\ \partial_\nu u = c & \text{ on $\Gamma$,} \end{array} \right.

where

\Omega

is a locally Lipschitz epigraph, that is

C^3

on

\Gamma\subseteq\partial\Omega

, with

\partial\Omega\setminus\Gamma

consisting in nonaccumulating, countably many points. We provide a geometric inequality that allows us to deduce geometric properties of the sets

\Omega

for which monotone solutions exist. In particular, if

\mathcal{C} \in \mathbb{R}^n

is a cone and either

n=2

or

n=3

and

f \ge 0

, then there exists no solution of

\left\{ \begin{array}{ll} \Delta u + f(u) = 0 & \text{ in $\mathcal{C}$,} \\ u > 0 & \text{ in $\mathcal{C}$,} \\ u = 0 & \text{ on $\partial\mathcal{C}$,} \\ \partial_\nu u = c & \text{ on $\partial\mathcal{C} \setminus \{0\}$.} \end{array} \right.

This answers a question raised by Juan Luis Vázquez.
Loewner chains in the unit disk
Authors: Contreras Manuel D. , Díaz-Madrigal Santiago , Gumenyuk Pavel
Published: December 2011

In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.] of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [Bracci, F., Contreras, M.D. and Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. To appear in J. Reine Angew. Math.]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.
A counterexample for the geometric traveling salesman problem in the Heisenberg group
Authors: Juillet Nicolas
Published: December 2011

We are interested in characterizing the compact sets of the Heisenberg group that are contained in a curve of finite length. Ferrari, Franchi and Pajot recently gave a sufficient condition for those sets, adapting a necessary and sufficient condition due to P. Jones in the Euclidean setting. We prove that this condition is not necessary.
Constant curvature foliations in asymptotically hyperbolic spaces
Authors: Mazzeo Rafe , Pacard Frank
Published: December 2011

Let

(M,g)

be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on

\partial M

and Weingarten foliations in some neighbourhood of infinity in

M

. We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant

\sigma_k

-curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for

g

and various properties of the foliation. Unlike other recent works in this area, by Rigger ([The foliation of asymptotically hyperbolic manifolds by surfaces of constant mean curvature (including the evolution equations and estimates). Manuscripta Math. 113 (2004), 403-421]) and Neves-Tian ([Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. Geom. Funct. Anal. 19 (2009), no.3, 910-942], [Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds. II. J. Reine Angew. Math. 641 (2010), 69-93]), we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition.
$L^2$ boundedness for maximal commutators with rough variable kernels
Authors: Chen Yanping , Ding Yong , Li Ran
Published: December 2011

For

b\in BMO(\mathbb{R}^n)

and

k\in\mathbb{N}

, the

k

-th order maximal commutator of the singular integral operator

T

with rough variable kernels is defined by

T^{\ast}_{b,k}f(x) = \sup_{\varepsilon > 0} \biggl| \int_{|x-y| > \varepsilon} \frac{\Omega(x,x-y)}{|x-y|^n} (b(x)-b(y))^{k} f(y) dy \biggl|.

In this paper the authors prove that the

k

-th order maximal commutator

T^{\ast}_{b,k}

is a bounded operator on

L^2(\mathbb{R}^n)

if

\Omega

satisfies the same conditions given by Calderón and Zygmund. Moreover, the

L^2

-boundedness of the

k

-th order commutator of the rough maximal operator

M_\Omega

with variable kernel, which is defined by

M_{\Omega;b,k}f(x) = \sup_{r > 0} \dfrac{1}{r^{n}} \int_{|x-y| < r} |\Omega(x,x-y)| |b(x)-b(y)|^{k} |f(y)| dy,

is also given here. These results obtained in this paper are substantial improvement and extension of some known results.
Quantitative uniqueness for second order elliptic operators with strongly singular coefficients
Authors: Lin Ching-Lung , Nakamura Gen , Wang Jenn-Nan
Published: December 2011

In this paper we study the local behavior of a solution to second order elliptic operators with sharp singular coefficients in lower order terms. One of the main results is the bound on the vanishing order of the solution, which is a quantitative estimate of the strong unique continuation property. Our proof relies on Carleman estimates with carefully chosen phases. A key strategy in the proof is to derive doubling inequalities via three-sphere inequalities. Our method can also be applied to certain elliptic systems with similar singular coefficients.
Submetries vs. submersions
Authors: Guijarro Luis , Walschap Gerard
Published: December 2011

We study submetries between Alexandrov spaces and show how some of the usual features of Riemannian submersions fail due to the lack of smoothness.
Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation
Authors: Cañizo José A. , Mischler Stéphane
Published: December 2011

We consider Smoluchowski's equation with a homogeneous kernel of the form

a(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha

with

-1 < \alpha \leq \beta < 1

and

\lambda := \alpha + \beta \in (-1,1)

. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at

y = 0

in the case

\alpha < 0

. We also give some partial uniqueness results for self-similar profiles: in the case

\alpha = 0

we prove that two profiles with the same mass and moment of order

\lambda

are necessarily equal, while in the case

\alpha < 0

we prove that two profiles with the same moments of order

\alpha

and

\beta

, and which are asymptotic at

y = 0

, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.
Nonnegative solutions of the heat equation on rotationally symmetric Riemannian manifolds and semismall perturbations
Authors: Murata Minoru
Published: December 2011

Let

M

be a rotationally symmetric Riemannian manifold, and

\Delta

be the Laplace-Beltrami operator on

M

. We establish a necessary and sufficient condition for the constant function 1 to be a semismall perturbation of

-\Delta +1

on

M

, and give optimal sufficient conditions for uniqueness of nonnegative solutions of the Cauchy problem to the heat equation. As an application, we determine the structure of all nonnegative solutions to the heat equation on

M\times(0,T)

.
Steiner and Schwarz symmetrization in warped products and fiber bundles with density
Authors: Morgan Frank , Howe Sean , Harman Nate
Published: December 2011

We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density.