Related papers: Talagrand's influence inequality revisited

An inequality by Samorodnitsky states that if $f : \mathbb{F}_2^n \to \mathbb{R}$ is a nonnegative boolean function, and $S \subseteq [n]$ is chosen by randomly including each coordinate with probability a certain $\lambda = \lambda(q,\rho)…

It is shown that if (X,||.||_X) is a Banach space with Rademacher type p \ge 1, then for every integer n there exists an even integer m < Cn^{2-1/p}log n (C is an absolute constant), such that for every f:Z_m^n --> X, \Avg_{x,\e}[||f(x+…

We characterize those pairs of weights $ \sigma $ on $ \mathbb{R}$ and $ \tau $ on $ \mathbb{C}_+$ for which the Cauchy transform $\mathsf{C}_{\sigma} f (z) \equiv \int_{\mathbb{R}} \frac {f(x)} {x-z} \; \sigma (dx)$, $ z\in \mathbb{C}_+$,…

In this paper we address the following question: given a measure $\mu$ on $\mathbb{R}^n$, does there exists a constant $C>0$ such that, for any $m$-dimensional subspace $H \subset \mathbb{R}^n$ and any convex body $K \subset \mathbb{R}^n$,…

We prove a new version of isoperimetric inequality: Given a positive real $m$, a Banach space $B$, a closed subset $Y$ of metric space $X$ and a continuous map $f:Y \rightarrow B$ with $f(Y)$ compact $$\inf_FHC_{m+1}(F(X))\leq…

We prove concentration inequalities for general functions of weakly dependent random variables satisfying the Dobrushin condition. In particular, we show Talagrand's convex distance inequality for this type of dependence. We apply our…

In this paper we deal with an invariant ergodic hyperbolic measure $\mu$ for a diffeomorphism $f,$ assuming that $f$ it is either $C^{1+\alpha}$ or $f$ is $C^1$ and the Oseledec splitting of $\mu$ is dominated. We show that this system…

Let $P_{\alpha} f(x,t)$ be the Caffarelli-Silvestre extension of a smooth function $f(x): \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}_+:=\mathbb{R}^n\times (0,\infty).$ The purpose of this article is twofold. Firstly, we want to characterize…

The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in…

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and assume that the Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$, and let $q\in[1,\infty)$ and $d\in(0,\infty)$.…

It is shown that if (X, ||.||_X) is a Banach space with Rademacher cotype q then for every integer n there exists an even integer m< n^{1+1/q}$ such that for every f:Z_m^n --> X we have $\sum_{j=1}^n \Avg_x [ ||f(x+ (m/2) e_j)-f(x) ||_X^q ]…

Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of all non-singleton balls $B$ we consider the hierarchical Laplacian $L=L_{C}$. Choosing a sequence $\{\varepsilon (B)\}$ of…

We investigate a Grothendieck-type inequality for pairs of Banach spaces $E,F$ assuming $E$ is finite-dimensional and study the associated Grothendieck-type constant. We prove that if there is a $C >0$ such that $\|A\otimes…

We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy,…

In this article we derive Talagrand's $T_2$ inequality on the path space w.r.t. the maximum norm for various stochastic processes, including solutions of one-dimensional stochastic differential equations with measurable drifts, backward…

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove…

We present several operator extensions of the Chebyshev inequality for Hilbert space operators. The main version deals with the synchronous Hadamard property for Hilbert space operators. Among other inequalities, it is shown that if…

The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1,1,...,1,...) is proved in a first course on functional…

Motivated by the result of Dantas et. al. (2023) that there exist metric spaces for which the set of strongly norm-attaining Lipschitz functions does not contain an isometric copy of $c_0$, we introduce and study a weaker notion of…

We consider a generic modified logarithmic Sobolev inequality (mLSI) of the form $\mathrm{Ent}_{\mu}(e^f) \le \tfrac{\rho}{2} \mathbb{E}_\mu e^f \Gamma(f)^2$ for some difference operator $\Gamma$, and show how it implies two-level…