Related papers: Maximal function estimates and self-improvement re…
A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal…
A comprehensive approach to Sobolev-type embeddings, involving arbitrary rearrangement- invariant norms on the entire Euclidean space R^n, is offered. In particular, the optimal target space in any such embedding is exhibited. Crucial in…
A sharp Poincar\'e-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso second-variation inequality. The…
Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $\mu.$ We establish a family of uniform Poincar\'e inequalities on $(X,d,\mu)$ if it satisfies a local Poincar\'e inequality ($P_{loc}$) and a condition on growth…
We make a study of Weinstein functionals, first defined in ~\cite{W}, on the hyperbolic space $\mathbb{H}^n$. We are primarily interested in the existence of Weinstein functional maximisers, or, in other words, existence of extremal…
We give an intrinsic characterization of the restrictions of Sobolev, Triebel-Lizorkin and Besov spaces to regular subsets of $R^n$ via sharp maximal functions and local approximations.
The aim of this paper is to establish various functional inequalities for the convolution of a compactly supported measure and a standard Gaussian distribution on Rd. We especially focus on getting good dependence of the constants on the…
In this work, we study the extremal functions of the log-Sobolev functional on compact metric measure spaces satisfying the $\mathrm{RCD}^*(K,N)$ condition for $K$ in $\mathbb{R}$ and $N$ in $(2,\infty)$. We show the existence, regularity…
In this paper, we carry out in-depth research centering around the $(p, q)$-Sobolev inequality and Nash inequality on forward complete Finsler metric measure manifolds under the condition that ${\rm Ric}_{\infty} \geq -K$ for some $K \geq…
A new modulus of smoothness and its equivalent $K$-function are defined on the conic domains in $\mathbb{R}^d$, and used to characterize the weighted best approximation by polynomials. Both direct and weak inverse theorems of the…
Given a smooth positive function $f$ defined on the unit circle satisfying a simple condition, we obtain a Poincar\'{e}-type inequality for an arbitrary function $u$ whose weighted average with respect to $f$ is zero. The proof uses…
In this paper, we prove interior Poincar{\'e} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike…
We provide an improvment of the maximum principle of Pon-tryagin of the optimal control problems, for a system governed by an ordinary differential equation, in presence of final constraints, in the setting of the piece-wise differentiable…
In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function…
Recent works of Guo-Phong-Song-Sturm established for compact K\"ahler manifolds (even for K\"ahler spaces of specific singularities) a variety of geometric estimates depending on an upper bound of $L^{1+\epsilon}$ or $L^1(\log…
We firstly describe a maximal inequality for dual Sobolev spaces W^{-1,p}. This one corresponds to a "Sobolev version" of usual properties of the Hardy-Littlewood maximal operator in Lebesgue spaces. Even in the euclidean space, this one…
We study in this article a new pointwise estimate for ''rough'' singular integral operators. From this pointwise estimate we will derive Sobolev type inequalities in a variety of functional spaces.
Eigenfunctions of the Schrodinger equation with the Coulomb potential in the imaginary Lobachevsky space are studied in two coordinate systems admitting solutions in terms of hypergeometric functions. Normalization and coefficients of…
The goal of this paper is to give the necessary and sufficient condition for Banach function spaces on which Young's inequality holds. As an application, we consider the maximal regularity estimate of heat equations for Besov spaces…
Poincar\'e inequality is a fundamental property that rises naturally in different branches of mathematics. The associated Poincar\'e constant plays a central role in many applications since it governs the convergence of various practical…