Related papers: A Simple Approach to Functional Inequalities for N…

We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincar\'e inequality with respect to surface measure is uniformly rectifiable.

In this paper, we establish a class of generalized Poincar\'{e}-type inequalities for torsional rigidity on the boundary of a convex body of class $C^{2}_{+}$ in $\rnnn$ by using the concavity of related Brunn-Minkowski inequality.

The quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O-U Dirichlet form on…

In the setting of a complete, doubling metric measure space $(X,d,\mu)$ supporting a $(1,1)$-Poincar\'e inequality, we show that for all $0<\theta<1$, the following fractional Poincar\'e inequality holds for all balls $B$ and locally…

We define the quantum $p$-divergences and introduce Beckner's inequalities for primitive quantum Markov semigroups on a finite-dimensional matrix algebra satisfying the detailed balance condition. Such inequalities quantify the convergence…

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with…

Let $G$ be a countable discrete group with an orthogonal representation $\alpha$ on a real Hilbert space $H$. We prove $L_p$ Poincar\'e inequalities for the group measure space $L_\infty(\Omega_H,\gamma)\rtimes G$, where both the group…

By introducing an intrinsic perimeter measure for intrinsic countably rectifiable sets, we prove a compactness result and a Poincar\'e inequality for special functions with bounded variation in equiregular Carnot-Carath\'eodory spaces which…

In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincare' inequality. We give a definition of parabolic De Giorgi classes and…

We give an extension of Poincar\'e's type capacitary inequality for Dirichlet spaces and provide an application to study the uniqueness sets on the unit circle for these spaces.

Explicit rate of convergence in variance (or more general entropies) is obtained for a class of Piecewise Deterministic Markov Processes such as the TCP process, relying on functional inequalities. A method to establish Poincar\'e (and more…

We obtain and study new $\Phi$-entropy inequalities for diffusion semigroups, with Poincar\'e or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear…

We present in a unified setting the foundations for a theory of non-bilinear Dirichlet functionals on Hilbert spaces. We prove known and new equivalences between non-linear semigroups, non-linear resolvents, non-linear generators, and their…

Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincar\'e inequality. We study extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ of $X$ and use them to…

We prove that synthetic lower Ricci bounds for metric measure spaces -- both in the sense of Bakry-\'Emery and in the sense of Lott-Sturm-Villani -- can be characterized by various functional inequalities including local Poincar\'e…

We prove estimates for the sharp constants in fractional Poincar\'e-Sobolev inequalities associated to an open set, in terms of a nonlocal capacitary extension of its inradius. This work builds upon previous results obtained in the local…

We employ a Markov semigroup approach combined with the $\Gamma$-calculus to establish a generalized Beckner inequality associated with weighted Gaussian measures. As a direct consequence, we derive the corresponding Poincar\'e inequality…

In this work we study a class of nonlocal quadratic forms given by \[ \mathcal{E}_j(u,v)=\frac{1}{2}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}(u(x)-u(y))(v(x)-v(y))j(x-y)\ dxdy, \] where $j:\mathbb{R}^N\to[0,\infty]$ is a measurable even…

In this paper, we extend the framework of Brezis--Van Schaftingen--Yung type inequalities in metric measure spaces by exploring several novel directions. First, we establish finite difference characterizations and fractional Sobolev-type…

We obtain improved fractional Poincar\'e inequalities in John domains of a metric space $(X, d)$ endowed with a doubling measure $\mu$ under some mild regularity conditions on the measure $\mu$. We also give sufficient conditions on a…