Related papers: Logarithmic Sobolev inequalities for mollified com…

In this paper we study several inequalities of log-Sobolev type for Dunkl operators. After proving an equivalent of the classical inequality for the usual Dunkl measure $\mu_k$, we also study a number of inequalities for probability…

In the paper we represent two examples which are based on the properties of discrete measures. In the first part of the paper we prove that for each probability measure $\mu$, $\operatorname{supp}{\mu}=[-1,1]$, which logarithmic potential…

We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative…

We provide a complete characterization of compactness of Sobolev embeddings of radially symmetric functions on the entire space $\mathbb{R}^n$ in the general framework of rearrangement-invariant function spaces. We avoid any unnecessary…

We provide a new characterization of the logarithmic Sobolev inequality.

Given $N$ i.i.d. samples from a probability measure $\mu$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $\mu_N \to \mu$ in the negative Sobolev space $W^{-\alpha, p}$. When $W^{-\alpha, p}$ contains point…

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $\gamma$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex…

By using the Ricci flow, we study local rigidity theorems regarding scalar curvature, isoperimetric constant and best constant of $L^2$ logarithmic Sobolev inequality. Precisely, we prove that if a metric $g$ on an open set $V$ in an…

For a closed connected surface with a metric g, we consider the regularized trace of the inverse of the Laplace-Beltrami operator. We minimize this on the class of smooth metrics conformal to g having the same area, and show that the…

Classically, the continuous-time Langevin diffusion converges exponentially fast to its stationary distribution $\pi$ under the sole assumption that $\pi$ satisfies a Poincar\'e inequality. Using this fact to provide guarantees for the…

For a regular (in a sense) mapping $v:\mathbb{R}^n \to \mathbb{R}^d$ we study the following problem: {\sl let $S$ be a subset of $m$-critical a set $\tilde Z_{v,m}=\{{\rm rank} \nabla v\le m\}$ and the equality $\mathcal{H}^\tau(S)=0$ (or…

Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such…

In this note we give a simple, dimension independent, proof of the logarithmic Sobolev inequality on the Heisenberg groups $H_n=\R^{2n+1}$ using the measure preserving transformations of the Brownian motion. We have corrected some serious…

We provide a mild sufficient condition for a probability measure on the real line to satisfy a modified log-Sobolev inequality for convex functions, interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux type…

We consider $n\times n$ real symmetric and hermitian random matrices $H_{n,m}$ equals the sum of a non-random matrix $H_{n}^{(0)}$ matrix and the sum of $m$ rank-one matrices determined by $m$ i.i.d. isotropic random vectors with…

We consider convergence properties of the long-term behaviors with respect to the coefficient of the stochastic term for a nonautonomous stochastic $p$-Laplacian lattice equation with multiplicative noise. First, the upper semi-continuity…

We prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifold with nonnegative sectional curvature of arbitrary dimension and codimension, while the ambient manifold needs to…

We survey some of the mechanisms used to prove that naturally defined sequences in combinatorics are log-concave. Among these mechanisms are Alexandrov's inequality for mixed discriminants, the Alexandrov Fenchel inequality for mixed…

The main aim of this work is to compare two Borel measures thorough their moment matrices using a new notion of smallest and largest generalized eigenvalues. With this approach we provide information in problems as the localization of the…

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,\mu)$. The embedding of the Newton-Morrey-Sobolev space into the H\"older space is obtained if $\mathscr{X}$ supports a weak…