Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities

We study the symmetry property of extremal functions to a family of weighted Sobolev inequalities due to Caffarelli-Kohn-Nirenberg. By using the moving plane method, we prove that all non-radial extremal functions are axially symmetric with respect to a line passing through the origin.

     

Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces

In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications.     

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.      

Polynomials,Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

This paper consists of three main parts. One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups. Despite the extensive research after Jerison's work [3] on Poincaré-type inequalities for Hörmander's vector fields over the years, our results given here even in the nonweighted case appear to be new. Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE's involving vector fields. The main tools to prove such inqualities are approximating the Sobolev functions by polynomials associated with the left invariant vector fields on ?. Some very usefull properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Finding the existence of such polynomials is the second main part of this paper. Main results of these two parts have been announced in the author's paper in Mathematical Research Letters [38].The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on (?,δ) domains. Some results of weighted Sobolev spaces are also given here. We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously. In particular, we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions. Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.     

Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature

We apply the method of [J. Demange, From porous media equation to generalized Sobolev inequalities on a Riemannian manifold, preprint, http://www.lsp.ups-tlse.fr/Fp/Demange/, 2004] and [J. Demange, Porous Media equation and Sobolev inequalities under negative curvature, preprint, http://www.lsp.ups-tlse.fr/Fp/Demange/, 2004], based on the curvature-dimension criterion and the study of Porous Media equation, to the case of a manifold M with strictly positive Ricci curvature. This gives a new way to prove classical Sobolev inequalities on M. Moreover, this enables to improve non-critical Sobolev inequalities as well. As an application, we study the rate of convergence of the solutions of the Porous Media equation to the equilibrium.     

Improved Sobolev inequalities and Muckenhoupt weights on stratified Lie groups

We study in this article the improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces and Besov spaces . When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in Rn and is classically obtained by a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p=1, these two techniques are not available anymore and following M. Ledoux (2003) [12], in Rn, we will treat here the critical case p=1 for general stratified Lie groups in a weighted functional space setting. Finally, we will go a step further with a new generalization of improved Sobolev inequalities using weak-type Sobolev spaces.     

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.     

Isoperimetric Inequalities on Compact Manifolds

We prove an isoperimetric inequality on compact Riemannian manifolds corresponding to the limit case of a scale of optimal Sobolev inequalities.     

Log-Sobolev inequalities and regions with exterior exponential cusps

We begin by studying certain semigroup estimates which are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for proving that such log-Sobolev inequalities hold for Euclidean regions which satisfy a particular Hardy-type inequality. Our main application is to show that domains which have exterior exponential cusps, and hence have no Sobolev embedding theorem, satisfy such heat kernel bounds provided the cusps are not too sharp. Finally, we consider a rotationally invariant domain with an exponentially sharp cusp and prove that ultracontractivity breaks down when the cusp becomes too sharp.     

A logarithmic Hardy inequality

We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue's measure nor a probability measure. All terms are scale invariant. After an Emden-Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved by radial functions, in some range of the parameters.     

Optimal Constants in Critical Sobolev Inequalities on Riemannian Manifolds in the Presence of Symmetries

We prove a theorem on the existence of a `second best constant' incritical Sobolev inequalities on compact Riemannian manifolds underthe action of an isometry group. The theorem is then applied toseveral examples initially introduced by different authors.     

On the equivalence of heat kernel estimates and logarithmic Sobolev inequalities for the Hodge Laplacian

In this paper we consider the Hodge Laplacian on differential k-forms over smooth open manifolds MN, not necessarily compact. We find sufficient conditions under which the existence of a family of logarithmic Sobolev inequalities for the Hodge Laplacian is equivalent to the ultracontractivity of its heat operator.We will also show how to obtain a logarithmic Sobolev inequality for the Hodge Laplacian when there exists one for the Laplacian on functions. In the particular case of Ricci curvature bounded below, we use the Gaussian type bound for the heat kernel of the Laplacian on functions in order to obtain a similar Gaussian type bound for the heat kernel of the Hodge Laplacian. This is done via logarithmic Sobolev inequalities and under the additional assumption that the volume of balls of radius one is uniformly bounded below.     

Multiple positive solutions for Robin problem involving critical weighted Hardy–Sobolev exponents with boundary singularities

This paper deals with the existence of positive solutions for Robin elliptic problems involving critical weighted Hardy–Sobolev exponents with boundary singularities. Using the Caffarelli–Kohn–Nirenberg inequalities and variational methods, we prove the existence and multiplicity of positive solutions.     

Weighted Lp‐theory for vector potential operators in three‐dimensional exterior domains

In the present paper, we study the vector potential problem in exterior domains of . Our approach is based on the use of weighted spaces in order to describe the behavior of functions at infinity. As a first step of the investigation, we prove important results on the Laplace equation in exterior domains with Dirichlet or Neumann boundary conditions. As a consequence of the obtained results on the vector potential problem, we establish useful results on weighted Sobolev inequalities and Helmholtz decompositions of weighted spaces. Copyright © 2015 John Wiley & Sons, Ltd.     

Quasielliptic operators and Sobolev type equations

We consider a class of matrix quasielliptic operators on the n-dimensional space. For these operators, we establish the isomorphism properties in some special scales of weighted Sobolev spaces. Basing on these properties, we prove the unique solvability of the initial value problem for a class of Sobolev type equations.     

Hardy and Rellich type inequalities on complete manifolds

We prove some Hardy and Rellich type inequalities on complete noncompact Riemannian manifolds supporting a weight function which is not very far from the distance function in the Euclidean space.     

Best constants for Sobolev inequalities for higher order fractional derivatives

We obtain sharp constants for Sobolev inequalities for higher order fractional derivatives. As an application, we give a new proof of a theorem of W. Beckner concerning conformally invariant higher-order differential operators on the sphere.     

Weighted inequalities for the Bochner-Riesz means related to the Fourier-Bessel expansions

We prove weighted inequalities for the Bochner-Riesz means for Fourier-Bessel series with more general weights w(x) than previously considered power weights. These estimates are given by using the local Ap theory and Hardy's inequalities with weights. Moreover, we also obtain weighted weak type (1,1) inequalities. The case when w(x)=xa is sketched and follows as a corollary of the main result.     

Taylor's inequalities in Orlicz–Sobolev type spaces

In this paper, we obtain inequalities involving the Taylor polynomial and weak derivatives of a function in an Orlicz–Sobolev type space. Moreover, we show that any such function can be expanded in a finite Taylor series almost everywhere. As a consequence, we prove that the coefficients of any extended best polynomial $L^{\mathrm{\Phi }}$-approximation of a function on a ball almost everywhere converge to the weak derivatives of such a function when the radius tends to 0. Lastly, we get a mean convergence result of such coefficients.