Related papers: Functional inequalities for Nonlocal Dirichlet For…

Let $V$ be a locally bounded measurable function such that $e^{-V}$ is bounded and belongs to $L^1(dx)$, and let $\mu_V(dx):=C_V e^{-V(x)} dx$ be a probability measure. We present the criterion for the weighted Poincar\'{e} inequality of…

Perturbations of super Poincar\'e and weak Poincar\'e inequalities for L\'evy type Dirichlet forms are studied. When the range of jumps is finite our results are natural extensions to the corresponding ones derived earlier for diffusion…

Let $V\in C^2(\R^d)$ such that $\mu_V(\d x):= \e^{-V(x)}\,\d x$ is a probability measure, and let $\aa\in (0,2)$. Explicit criteria are presented for the $\aa$-stable-like Dirichlet form $$\E_{\aa,V}(f,f):= \int_{\R^d\times\R^d}…

With direct and simple proofs, we establish Poincar\'{e} type inequalities (including Poincar\'{e} inequalities, weak Poincar\'{e} inequalities and super Poincar\'{e} inequalities), entropy inequalities and Beckner-type inequalities for…

We consider the symmetric non-local Dirichlet form $(E, F)$ given by \[ E (f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $F$ the closure of the set of $C^1$ functions on $R^d$ with compact support with respect to $E_1$,…

This paper is concerned with a class of singular stable-like Dirichlet forms on $\R^d$, which are generated by $d$ independent copies of a one-dimensional symmetric $\alpha$-stable process, and whose L\'evy jump kernel measure is…

Let $(E,\F,\mu)$ be a $\si$-finite measure space. For a non-negative symmetric measure $J(\d x, \d y):=J(x,y) \,\mu(\d x)\,\mu(\d y)$ on $E\times E,$ consider the quadratic form $$\E(f,f):= \frac{1}{2}\int_{E\times E} (f(x)-f(y))^2 \, J(\d…

Given a subset $D$ of the Euclidean space, we study nonlocal quadratic forms that take into account tuples $(x,y) \in D \times D$ if and only if the line segment between $x$ and $y$ is contained in $D$. We discuss regularity of the…

Consider the symmetric non-local Dirichlet form $(D,\D(D))$ given by $$ D(f,f)=\int_{\R^d}\int_{\R^d}\big(f(x)-f(y)\big)^2 J(x,y)\,dx\,dy $$with $\D(D)$ the closure of the set of $C^1$ functions on $\R^d$ with compact support under the norm…

In this paper we firstly derive the weak elliptic Harnack inequality from the generalized capacity condition, the tail estimate of jump measure and the Poincar\'{e} inequality, for any regular Dirichlet form without killing part on a…

We study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. We show that the limits of these nonlocal functionals are…

The paper provides two versions of nonlocal Poincar\'e-type inequalities for integral operators with a convolution-type structure and functions satisfying a zero-Dirichlet like condition. The inequalities extend existing results to a large…

Probability measures satisfying a Poincar{\'e} inequality are known to enjoy a dimension free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincar{\'e} inequality automatically…

We establish new sharp inequalities of Poincar\'{e} or log-Sobolev type, on geodesically-convex weighted Riemannian manifolds $(M,\mathfrak{g},\mu)$ whose (generalized) Ricci curvature $Ric_{\mathfrak{g},\mu,N}$ with effective dimension…

Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincar\'e inequality, have been handy for estimating the…

We propose a general approach for quantitative convergence analysis of non-reversible Markov processes, based on the concept of second-order lifts and a variational approach to hypocoercivity. To this end, we introduce the flow Poincar{\'e}…

In the context of metric measure spaces, we introduce an axiomatic formulation of mixed local and nonlocal $p$-energy forms. Within this framework, we use the Poincar\'{e} inequality, the cutoff Sobolev inequality, and mild assumptions on…

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. Compared with previous works, we consider more general functionals.…

Extremal functions are exhibited in Poincar\'e trace inequalities for functions of bounded variation in the unit ball ${\mathbb B}^n$ of the $n$-dimensional Euclidean space ${\mathbb R}^n$. Trial functions are subject to either a vanishing…

We establish transience criteria for symmetric non-local Dirichlet forms on $L^2({\mathbb R}^d)$ in terms of the coefficient growth rates at infinity. Applying these criteria, we find a necessary and sufficient condition for recurrence of…