normalisation holomorphe d algebres de type cartan de champs de vecteurs holomorphes singuliers nous montrons un resultat de normalisation holomorphe d une famille commutative de champs de vecteurs holomorphes au voisinage de leur point singulier commun pour ce faire nous supposons qu une condition diophanti enne portant sur une algebre de lie commutative de champs lineaires associee est satisfaite d autre part nous imposons certaines conditions algebriques sur leur forme normale formelle les champs d une telle famille sauf un peuvent ne pas avoir de partie lineaire a l origine abstract we consider a commutative family of holomorphic vector fields in a neigh bourhood of a common singular point say 0 e cn let g be a commuta tive complex lie algebra of dimension 1 let a1 an e and let us set s g l ai g xi we assume that this lie morphism is diophantine in some sense let x1 be a holomorphic vector fields in a neighbourhood of 0 e cn we assume that its linear part s is regular relative to s that it belongs to s g and has the same formal centralizer as s let x2 xi be holomorphic vector fields vanishing at 0 and commuting with x1 then there exists a formal diffeomorphism of cn o such that the family of vector fields are in normal form in these formal coordinates this means that each element of the family commutes with s we assume that the normal forms of the xi s belong to 0s 0 s g os is the ring of formal first integral of s we also assume that the family of lowest order parts of the normal forms is free over 9s then we show that there exists a holomorphic diffeomorphism of cn 0 which transforms the family into a normal form the elements of the family

sharp constants for moser trudinger inequalities on spheres in complex space ℂn the main results of this paper concern sharp constants for the moser trudinger inequalities on spheres in complex space c n we derive moser trudinger inequalities for smooth functions and holomorphic functions with different sharp constants see theorem 1 1 the sharp moser trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the cr setting though there is a close connection in spirit between inequalities proven here on complex spheres and those on the heisenberg group for functions with compact support in any finite domain proven earlier by the same authors 17 derivation of the sharp constants for moser trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the heisenberg group variants of moser onofri type inequalities are also given on complex spheres as applications of our sharp inequalities see theorems 1 2 and 1 3 one of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients see theorem 1 4 c 2004 wiley periodicals inc

fehlerabschatzungen fur gauβ quadraturformeln we consider gauss quadrature formulaeq n n approximating the integral i f int limits 1 1 w x f x dx w an even weight function let f z sum limits k 0 infty alpha k f z k be analytic ink r z z 1 and f r mathop sup limits k geqq n alpha 2k f r 2k infty the error functionalr n i q n is continuous with respect to r and the relation parallel r n parallel sum limits k 0 infty r n q 2k r 2k q2k x x 2k holds in this paper estimates for r n are given to this end we first derive two new representations of r n which are essential for our further investigations the r n r 2 r n with x 1 r 2 x 2 is estimated in various ways by using the best uniform approximation of in p2n 1 and also the expansion of with respect to chebyshe polynomials of the first and second kind forw x 1 x 2 1 2 r n is calculated the asymptotic behaviour forr 1 of r n and of the derived error bounds is also discussed finally we compare different error bounds and give numerical examples

on maps between nonminimal hypersurfaces we construct an analytic jet parametrization for nontrivial cr maps between real analytic 1 nonminimal hypersurfaces in mathbb c 2 in particular this gives a real analytic structure on the set of all such maps and shows that the automorphism groups of a 1 nonminimal hypersurface in mathbb c 2 is a lie group our results also hold for some classes of 1 nonminimal hypersurfaces in higher dimensions

a note on minimizing the moser trudinger functional of high order derivatives abstract let be a bounded smooth domain in r 2 n n 2 in this note we consider the functional j u 1 2 n 2 u x 2 d x log h x e u x d x suppose h x is a smooth function with 0 a h x b then using the idea of lin and wei c lin j wei locating the peaks of solutions via the maximum principle ii a local version of the method of moving planes comm pure appl math lvi 2003 784 809 we prove the existence of minimizers of j for any 2 n 2 2 n n n 1 2 n in a space of functions h h n u j u h 0 1 j 1 2 n 1 2 where 2 n is the area of the unit sphere s 2 n 1 in r 2 n

maximization problem on trudinger moser inequality involving lebesgue norm abstract we study a maximization problem on the trudinger moser inequality and clarify the effect of lower order perturbation on the attainability of the best constant in particular we show the explicit forms of threshold nonlinearity involving lower order perturbation our results are based on the estimates of the dirichlet energy of maximizers

local analytic reduction of families of diffeomorphisms abstract we study local analytic simplification of families of analytic maps near a hyperbolic fixed point a particularly important application of the main result concerns families of hyperbolic saddles where siegel s theorem is too fragile at least in the analytic category by relaxing on the formal normal form we obtain analytic conjugacies since we consider families it is more convenient to state some results for analytic maps on a banach space this gives no extra complications as an example we treat a family passing through a 1 1 resonant saddle

global compactness properties of semilinear elliptic equations with critical exponential growth abstract sequences of positive solutions to semilinear elliptic equations of critical exponential growth in the plane either are precompact in the sobolev h 1 topology or concentrate at isolated points of the domain for energies allowing at most single point blow up we establish a universal blow up pattern near the concentration point and uniquely characterize the blow up energy in terms of a geometric limiting problem

improved moser trudinger type inequalities in the hyperbolic space hn we establish an improved version of the moser trudinger inequality in the hyperbolic space hnhn n 2n 2 namely we prove the following result for any 0 n 1nn then we have supu c0 hn hn gu gndvolg hn u ndvolg 1 hn n n u nn 1 dvolg where n n n 11n 1 n 1 n 1 denotes the surface area of the unit sphere in rnrn and n t et j 0n 2tjj this improves the moser trudinger inequality in hyperbolic spaces obtained recently by mancini and sandeep 2010 by mancini et al 2013 and by adimurthi and tintarev 2010 for 0 0 in the limiting case n 1n n we prove a moser trudinger inequality with exact growth in hnhn supu c0 hn hn gu gndvolg n 1n n hn u ndvolg 11 hn u ndvolg hn n n u nn 1 1 u nn 1dvolg this improves the moser trudinger inequality with exact growth in hnhn established by lu and tang 2016 these inequalities are achieved from the comparison of the symmetric non increasing rearrangement of a function both in the hyperbolic and the euclidean spaces and the same inequalities in the euclidean space this approach seems to be new comparing with the previous ones

extremals for moser inequalities this paper shows that extremals exist for four moser inequalities involving exponential integrals the results are for sobolev spaces defined on the spheres s2 and s3 on the disk d2 and on rp2 the method that carleson and chang used for the n ball with boundary value zero can be modified and applied to functions with mean value zero on these domains this approach also provides an elementary proof of the moser inequality for sn n 2 which is a special case of a result of fontana

on an n manifold in mathbf c n near an elliptic complex tangent in this paper we will be concerned with the local biholomorphic properties of a real n manifold m in c at a generic point such a manifold basically has the nature of the standard r in c near a complex tangent however the consideration can be much more complicated and the manifold may acquire a nontrivial local hull of holomorphy and many other biholomorphic invariants the study of such a problem was first carried out in a celebrated paper of e bishop bis where for each sufficiently non degenerate complex tangent he attached a biholomorphic invariant called the bishop invariant when the complex tangent is elliptic i e when 0 12 for a more precise definition see 2 he showed the existence of families of complex analytic disks with boundary on m that shrink down to the locus of points in m with complex tangents in particular using the well known continuity principle one sees that the image m of such families is contained in the holomorphic hull of the manifold at the time he asked whether m gives precisely the local holomorphic hull of m as well as certain uniqueness properties of the attached disks he also proposed the problem of determining the fine structure of m near such complex tangents later there appeared a sequence of papers concerning the smooth character of m in case m c here we would like to mention in particular the famous theorem proved by kenig webster in their deep work kw1 which states that the local hull of holomorphy m near an elliptic complex tangent is a smooth levi flat hypersurface with m c as part of its smooth boundary in another important paper of moser webster mw a systematic normal form theory was employed for the understanding of the local biholomorphic invariants of m in case m is real analytic when the bishop invariant 6 0 their method works in any dimension and even for some hyperbolic complex tangents but it breaks down for complex tangents with 0 among other things they showed that m can be biholomorphically mapped into the affine space

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

maximizers for the variational problems associated with sobolev type inequalities under constraints we propose a new approach to study the existence and non existence of maximizers for the variational problems associated with sobolev type inequalities both in the subcritical case and critical case under the equivalent constraints the method is based on an useful link between the attainability of the supremum in our variational problems and the attainablity of the supremum of some special functions defined on 0 infty our approach can be applied to the same problems related to the fractional laplacian operators our main results are new in the critical case and in the setting of the fractional laplacian operator which was left open in the work of ishiwata and wadade math ann 364 1043 1068 2016 and ishiwata on variational problems associated with sobolev type inequalities and related topics http www rism it doc ishiwata pdf 2015 in the subcritical case our approach provides a new and elementary proof of the results of ishiwata and wadade 2016 and ishiwata 2015

flattening of cr singular points and analyticity of the local hull of holomorphy i this is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in mathbb c n with n ge 3 near a non degenerate cr singular point the problem is motivated from the study of the complex plateau problem that seeks for the levi flat hypersurface bounded by a given real submanifold and is motivated by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space the present article is focused on the case of cr singular points with at least one elliptic direction we solve the holomorphic flattening problem and thus provide a complete description of the local hull of holomorphy in this setting the results in this paper and those in flattening of cr singular points and analyticity of the local hull of holomorphy ii p 60 2014 are taken from our arxiv post flattening of cr singular points and analyticity of the local hull of holomorphy 2012 we split flattening of cr singular points and analyticity of the local hull of holomorphy 2012 into two independent articles to avoid it being too long

moser trudinger trace inequalities sharp constants in exponential inequalities involving a general class of measures in domains are exhibited in the limiting case of the sobolev embedding theorem a comprehensive approach is presented yielding as special instances trace inequalities on o on smooth submanifolds of o of arbitrary dimension and also on fractal subsets of o and recovering in particular the classical moser trudinger inequality

sharp moser trudinger inequality on the heisenberg group at the critical case and applications let h cn r be the n dimensional heisenberg group q 2n 2 be the homogeneous dimension of h q qq 1 and z 4 t2 14 be the homogeneous norm of z t h then we prove the following sharp moser trudinger inequality on h theorem 1 6 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q 0 q 1 q there holds sup u 1 1 h1 exp u q q 1 k 0q 2 kk u kq q 1 c q the constant q 1 q is best possible in the sense that the supremum is infinite if q 1 q here is any positive number and u 1 h hu q h u q 1 q our result extends the sharp moser trudinger inequality by cohn and lu 2001 19 on domains of finite measure on h and sharpens the recent result of cohn et al 2012 18 where such an inequality was studied for the subcritical case q 1 q we carry out a completely different and much simpler argument than that in cohn et al 2012 18 to conclude the critical case our method avoids using the rearrangement argument which is not available in an optimal way on the heisenberg group and can be used in more general settings such as riemanian manifolds appropriate metric spaces etc as applications we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of q laplacian type on the heisenberg group theorems 1 8 1 9 1 10 and 1 11 divh hu q 2 hu v u q 2u f u eh with nonlinear terms f of maximal exponential growth exp u qq 1 as u in particular when e 0 the existence of a nontrivial solution is also given

extremal functions for moser s inequality let q be a bounded smooth domain in ri and u x a c1 function with compact support in q moser s inequality states that there is a constant co depending only on the dimension n such that 1 wn1 n j ee n u dx co where iqi is the lebesgue measure of q and wn 1 the surface area of the unit ball in rn we prove in this paper that there are extremal functions for this inequality in other words we show that the sup j nw u dx u e wj 11vuh1n 1 is attained earlier results include carleson chang 1986 q is a ball in any dimension and flucher 1992 q is any domain in 2 dimensions

isoperimetry for semilinear torsion problems in riemannian two manifolds abstract given m 2 g a smooth complete riemannian 2 manifold this note discusses three types of isoperimetric properties arising from u z o g o g z u d a g the non negative solution to the semilinear 0 1 torsion problem g u u in o subject to u o 0 for a relatively compact domain o m 2 with smooth boundary o

a normal form for a real 2 codimensional submanifold in cn 1 near a cr singularity abstract we construct a formal normal form for a class of real submanifolds m c n 1 of codimension 2 near a cr singularity approximating the sphere our result gives a generalization of huang yin s normal form in c 2 to the higher dimensional analog case

flattening of cr singular points and analyticity of the local hull of holomorphy ii abstract this is the second article of the two papers in which we investigate the holomorphic and formal flattening problem of a non degenerate cr singular point of a codimension two real submanifold in c n with n 3 the problem is motivated from the study of the complex plateau problem that looks for the levi flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space the present article is focused on non degenerate flat cr singular points with at least one non parabolic bishop invariant we will solve the formal flattening problem in this setting the results in this paper and those in 23 are taken from our earlier arxiv post 22 we split 22 into two independent articles to avoid it being too long

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

blow up analysis concerning singular trudinger moser inequalities in dimension two abstract in this paper we derive a sharp version of the singular trudinger moser inequality which was originally established by adimurthi and sandeep 2 moreover extremal functions for those singular trudinger moser inequalities are also obtained our method is the blow up analysis compared with our previous work 32 the essential difficulty caused by the presence of singularity is how to analyse the asymptotic behavior of certain maximizing sequence near the blow up point we overcome this difficulty by combining two different classification theorems of chen and li 6 7 to get the desired bubble

adams inequalities on measure spaces abstract in 1988 adams obtained sharp moser trudinger inequalities on bounded domains of r 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

existence and nonexistence of extremal functions for sharp trudinger moser inequalities abstract our main purpose in this paper is to establish the existence and nonexistence of extremal functions also known as maximizers and symmetry of extremals for several trudinger moser type inequalities on the entire space r n including both the critical and subcritical trudinger moser inequalities see theorems 1 1 1 2 1 3 1 4 1 5 most of earlier works on existence of maximizers in the literature rely on the complicated blow up analysis of pdes for the associated euler lagrange equations of the corresponding moser functionals the new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical trudinger moser inequalities and the critical ones established by the same authors in 25 combining with the continuity of the supremum function that is observed for the first time in the literature these allow us to establish the existence and nonexistence of the maximizers for the trudinger moser inequalities in different ranges of the parameters including those inequalities with the exact growth this method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular truddinger moser inequalities are symmetric moreover we will be able to calculate the exact values of the supremums of the trudinger moser type in certain cases these appear to be the first results in this direction

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

geometrical logarithmic capacitance abstract this paper is devoted to a novel and nontrivial exploration of eight aspects of the geometrical logarithmic capacitance a very key notion in mathematical physics quasiconformal geometry and variational calculus through 1 identifying with the reduced conformal module 2 evaluating the minimal log potential energy 3 relating to both the volume radius and the surface radius 4 linking with the n harmonic radius and the log capacity of the kevin image of a compact surface 5 finding the minkowski inequality and the general variational formula for the log capacity 6 pinching the log isocapacitary inequality from left and solving the left prescribed problem for the normalized log capacitary curvature measure 7 pinching the log isocapacitary inequality from right and handling the right prescribed problem for the normalized log capacitary curvature measure 8 handling an overdetermination of the n equilibrium potential of a given convex body via the log capacitary concavity

a weighted form of moser trudinger inequality on riemannian surface abstract let g be a compact riemannian surface without boundary in this paper we shall use blowing up analysis to prove that sup u 2 f d v g 1 u d v g 0 e 4 f u 2 d v g furthermore we show that there exists an extremal function for the above inequality in other words we show that sup u 2 f d v g 1 u d v g 0 e 4 f u 2 d v g is attained

extremal functions for trudinger moser inequalities of adimurthi druet type in dimension two abstract combining carleson chang s result 9 with blow up analysis we prove existence of extremal functions for certain trudinger moser inequalities in dimension two this kind of inequality was originally proposed by adimurthi and o druet 1 extended by the author to high dimensional case and riemannian surface case 40 41 generalized by c tintarev to wider cases including singular form 36 and by m de souza and j m do o 14 to the whole euclidean space r 2 in addition to the euclidean case we also consider the riemannian surface case the results in the current paper complement that of l carleson and a chang 9 m struwe 35 m flucher 16 k lin 19 and adimurthi and o druet 1 our previous ones 26 41 and part of c tintarev 36

the error norm of gaussian quadrature formulae for weight functions of bernstein szego type we consider the gauss kronrod quadrature formulae for the bernstein szego weight functions consisting of any one of the four chebyshev weights divided by the polynomial on certain spaces of analytic functions the error term of these formulae is a continuous linear functional we compute explicitly the norm of the error functional

trudinger moser type inequalities with logarithmic weights in dimension n abstract we consider borderline embeddings of trudinger moser type for weighted sobolev spaces in bounded domains in r n the embeddings go into orlicz spaces with exponential growth functions it turns out that the most interesting weights are powers of the logarithm for which an explicit dependence of the maximal growth functions can be established corresponding moser type results are also proved with explicit sharp exponents in the particular case of a logarithmic weight with the limiting exponent n 1 a maximal growth of double exponential type is obtained while for any larger exponent the embedding goes into l

real submanifolds of maximum complex tangent space at a cr singular point i we study a germ of real analytic n dimensional submanifold of mathbf c n that has a complex tangent space of maximal dimension at a cr singularity under some assumptions we show its equivalence to a normal form under a local biholomorphism at the singularity we also show that if a real submanifold is formally equivalent to a quadric it is actually holomorphically equivalent to it if a small divisors condition is satisfied finally we investigate the existence of a complex submanifold of positive dimension in mathbf c n that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a possibly non isolated cr singularity

on a sharp sobolev type inequality on two dimensional compact manifolds motivated by the asymptotic analysis of double vortex condensates in the chern simons higgs theory we construct a suitable minimizing sequence for a sharp sobolev inequality a la moser for two dimensional compact manifolds as a consequence we first obtain a direct proof of the sharp character of such an inequality secondly and more interestingly we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum

the sharp sobolev type inequalities in the lorentz sobolev spaces in the hyperbolic spaces abstract let w 1 l p q h n 1 q p denote the lorentz sobolev spaces of order one in the hyperbolic spaces h n our aim in this paper is three fold first of all we establish a sharp poincare inequality in w 1 l p q h n with 1 q p which generalizes the result in 41 to the setting of lorentz sobolev spaces second we prove several sharp poincare sobolev type inequalities in w 1 l p q h n with 1 q p n which generalize the results in 45 to the setting of lorentz sobolev spaces finally we provide the improved moser trudinger type inequalities in w 1 l n q h n in the critical case p n with 1 q n which generalize the results in 43 and improve the results in 57 in the proof of the main results we shall prove a polya szego type principle in w 1 l p q h n with 1 q p which maybe is of independent interest

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

moser inequalities in gauss space the sharp constants in a family of exponential sobolev type inequalities in gauss space are exhibited they constitute the gaussian analogues of the moser inequality in the borderline case of the sobolev embedding in the euclidean space interestingly the gaussian results have features in common with the euclidean ones but also reveal marked diversities

the thresholds of the existence of maximizers for the critical sharp singular moser trudinger inequality under constraints this paper is addressed to study the existence of maximizers for the singular moser trudinger supremum under constraints in the critical case begin aligned mt n a beta sup u in w 1 n mathbb r n vert nabla u vert n a vert u vert n n 1 int mathbb r n phi n left 1 beta n alpha n u frac n n 1 right x beta dx end aligned where a 0 beta in 0 n phi n t e t sum k 0 n 2 frac t k k alpha n n omega n 1 1 n 1 and omega n 1 denotes the surface area of the unit sphere in mathbb r n more precisely we study the effect of the parameter a to the attainability of mt n a beta we will prove that for each beta in 0 n there exist the thresholds a beta and a beta such that mt n a beta is attained for any a in a beta a beta and is not attained for a a beta we also give some qualitative estimates for a beta and a beta our results complete the recent studies on the sharp moser trudinger type inequality under constraints due to do o sani and tarsi commun contemp math 19 27 2016 lam proc am math soc 145 4885 4892 2017 math nachr 291 14 15 2272 2287 2018 and ikoma ishiwata and wadade math ann 373 1 2 831 851 2019

analytic aspects of the toda system i a moser trudinger inequality in this paper we analyze solutions of the open toda system and establish an optimal moser trudinger type inequality for this system let be a closed surface with area 1 and k aij n n the cartan matrix for su n 1 i e we show that has a lower bound in h1 n if and only if this inequality is optimal as a direct consequence if mj for 4 for j 1 2 n m has a minimizer u that satisfies 2001 john wiley sons inc

a sharp form of the moser trudinger inequality on a compact riemannian surface in this paper a sharp form of the moser trudinger inequality is established on a compact riemannian surface via the method of blow up analysis and the existence of an extremal function for such an inequality is proved

trudinger moser inequalities on complete noncompact riemannian manifolds abstract let m g be a complete noncompact riemannian n manifold n 2 if there exist positive constants and such that sup u w 1 n m u 1 1 m e u n n 1 k 0 n 2 k u n k n 1 k d v g where u 1 g u l n m u l n m then we say that the trudinger moser inequality holds suppose the trudinger moser inequality holds we prove that there exists some positive constant such that vol g b x 1 for all x m also we give a sufficient condition under which the trudinger moser inequality holds say the ricci curvature of m g has lower bound and its injectivity radius is positive moreover the adams inequality is discussed in this paper for application of the trudinger moser inequality we obtain existence results for some quasilinear equations with nonlinearity of exponential growth

a hardy moser trudinger inequality abstract in this paper we obtain an inequality on the unit disk b in r 2 which improves the classical moser trudinger inequality and the classical hardy inequality at the same time namely there exists a constant c 0 0 such that b e 4 u 2 h u d x c 0 u c 0 b 0 where h u b u 2 d x b u 2 1 x 2 2 d x this inequality is a two dimensional analog of the hardy sobolev maz ya inequality in higher dimensions which has been intensively studied recently we also prove that the supremum is achieved in a suitable function space which is an analog of the celebrated result of carleson chang for the moser trudinger inequality

on the effect of equivalent constraints on a maximizing problem associated with the sobolev type embeddings in mathbb r n in this paper we consider the attainability of a maximizing problem begin aligned d sup vert u vert h 1 n gamma 1 left vert u vert n n alpha vert u vert p p right end aligned where n ge 2 n 0 0 gamma le n and vert u vert h 1 n gamma left vert u vert n gamma vert nabla u vert n gamma right frac 1 gamma the existence of a maximizer for d is closely related to the exponent gamma in fact we show that the value begin aligned alpha alpha inf vert u vert h 1 n gamma 1 left frac 1 vert u vert n n vert u vert p p right end aligned is a threshold in terms of the attainability of d

a double mean field equation related to a curvature prescription problem abstract we study a double mean field type pde related to a prescribed curvature problem on compacts surfaces with boundary 0 1 u 2 k e u k e u 1 in u 2 h e u 2 h e u 2 1 on here and are real parameters k h are smooth positive functions on and respectively and is the outward unit normal vector to we provide a general blow up analysis then a moser trudinger inequality which gives energy minimizing solutions for some range of parameters finally we provide existence of min max solutions for a wider range of parameters which is dense in the plane if is not simply connected

convergence of the chern moser beloshapka normal forms in this article we first describe a normal form of real analytic levi nondegenerate submanifolds of c n of codimension d 1 under the action of formal biholomorphisms that is of perturbations of levi nondegenerate hyperquadrics we give a sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic we show that our techniques can be adapted in the case d 1 in order to obtain a new and direct proof of chern moser normal form theorem

a sharp form of trace moser trudinger inequality on compact riemannian surface with boundary in this paper a sharp form of trace moser trudinger inequality is established on the boundary of compact riemannian surface and the existence of extremal function is proved via the method of blowing up analysis

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

convergence and finite determination of formal cr mappings in this paper we study the convergence and finite determination of formal holomorphic mappings of c 0 taking one real submanifold into another by a formal holomorphic mapping h c 0 c 0 we mean an n vector h h1 hn where each hj is a formal power series in n indeterminates with no constant term if m and m are real smooth submanifolds through 0 in c defined near the origin by z z 0 and z z 0 respectively where and are vector valued smooth defining functions then we say that a formal mapping h c 0 c 0 sends m into m if the vector valued power series h z h z is a matrix multiple of z z for real analytic hypersurfaces we shall prove the following theorem 1 let m and m be real analytic hypersurfaces through the origin in c n 2 assume that neither m nor m contains a nontrivial holomorphic subvariety through 0 then any formal mapping h c 0 c 0 sending m into m is convergent the condition that m above does not contain a nontrivial holomorphic subvariety is necessary see remark 2 3 as a corollary we obtain the following characterization

existence and nonexistence of extremals for critical adams inequalities in r4 and trudinger moser inequalities in r2 abstract though much progress has been made with respect to the existence of extremals of the critical first order trudinger moser inequalities in w 1 n r n and higher order adams inequalities on finite domain r n whether there exists an extremal function for the critical higher order adams inequalities on the entire space r n still remains open the current paper represents the first attempt in this direction by considering the critical second order adams inequality in the entire space r 4 the classical blow up procedure cannot apply to solving the existence of critical adams type inequality because of the absence of the polya szego type inequality in this paper we develop some new ideas and approaches based on a sharp fourier rearrangement principle see 31 sharp constants of the higher order gagliardo nirenberg inequalities and optimal poly harmonic truncations to study the existence and nonexistence of the maximizers for the adams inequalities in r 4 of the form s sup u h 2 1 r 4 exp 32 2 u 2 1 u 2 d x where 32 2 we establish the existence of the threshold where 32 2 2 b 2 2 and b 2 1 24 2 such that s is attained if 32 2 and is not attained if 32 2 this phenomenon has not been observed before even in the case of first order trudinger moser inequality therefore we also establish the existence and non existence of an extremal function for the trudinger moser inequality on r 2 furthermore the symmetry of the extremal functions can also be deduced through the fourier rearrangement principle

existence and non existence of maximizers for the moser trudinger type inequalities under inhomogeneous constraints in this paper we study the existence and non existence of maximizers for the moser trudinger type inequalities in mathbb r n of the form begin aligned d n alpha a b sup u in w 1 n mathbb r n vert nabla u vert l n mathbb r n a vert u vert l n mathbb r n b 1 int mathbb r n phi n left alpha u n right dx end aligned here n ge 2 n frac n n 1 a b 0 alpha in 0 alpha n and phi n t e t sum j 0 n 2 frac t j j where alpha n n omega n 1 1 n 1 and omega n 1 denotes the surface area of the unit ball in mathbb r n we show the existence of the threshold alpha alpha a b n in 0 alpha n such that d n alpha a b is not attained if alpha in 0 alpha and is attained if alpha in alpha alpha n we also provide the conditions on a b in order that the inequality alpha alpha n holds

convergence or generic divergence of the birkhoff normal form we prove that the birkhoff normal form of hamiltorlian flows at a nonresonant singular point with given quadratic part is always convergent or generically divergent the same result is proved for the normalization mapping and any formal first integral

convergence and divergence of formal cr mappings let m subset mathbb c n be a generic real analytic submanifold of finite type m subset mathbb c n be a real analytic set and p in m where we assume that n n geqslant 2 let h mathbb c n p to mathbb c n be a formal holomorphic mapping sending m into m and let mathcal e m denote the set of points in m through which there passes a complex analytic subvariety of positive dimension contained in m we show that if h does not send m into mathcal e m then h must be convergent as a consequence we derive the convergence of all formal holomorphic mappings when m does not contain any complex analytic subvariety of positive dimension answering by this a long standing open question in the field more generally we establish necessary conditions for the existence of divergent formal maps even when the target real analytic set is foliated by complex analytic subvarieties allowing us to settle additional convergence problems such as e g for transversal formal maps between levi non degenerate hypersurfaces and for formal maps with range in the tube over the light cone

a bishop surface with a vanishing bishop invariant we give a solution to the equivalence problem for bishop surfaces with the bishop invariant 0 as a consequence we answer in the negative a problem that moser asked in 1985 after his work with webster in 1983 and his own work in 1985 this will be done in two major steps we first derive the formal normal form for such surfaces we then show that two real analytic bishop surfaces with 0 are holomorphically equivalent if and only if they have the same formal normal form up to a trivial rotation our normal form is constructed by an induction procedure through a completely new weighting system from what is used in the literature our convergence proof is done through a new hyperbolic geometry associated with the surface