Related papers: The One Dimensional Free Poincar\'e Inequality

We prove a sharp Poincar\'e inequality for subsets $\Omega$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This is achieved by…

It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is an Abelian integral also in non generic cases. In non generic cases it is an…

Let (N, G), where N is a normal subgroup of G<SL_n(C), be a pair of finite groups and V a finite-dimensional fundamental G-module. We study the G-invariants in the symmetric algebra S(V) by giving explicit formulas of the Poincar\'{e}…

A tensor extension of the Poincar\'e algebra is proposed for the arbitrary dimensions. Casimir operators of the extension are constructed. A possible supersymmetric generalization of this extension is also found in the dimensions $D=2,3,4$.

We prove stability estimates for the Bakry-Emery bound on Poincar\'e and logarithmic Sobolev constants of uniformly log-concave measures. In particular, we improve the quantitative bound in a result of De Philippis and Figalli asserting…

We prove a Poincar\'e-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical…

This work studies mixtures of probability measures on $\mathbb{R}^n$ and gives bounds on the Poincar\'e and the log-Sobolev constant of two-component mixtures provided that each component satisfies the functional inequality, and both…

We deal with regular Lagrangian constrained systems which are invariant under the action of a symmetry group. Fixing a connection on the higher-order principal bundle where the Lagrangian and the (independent) constraints are defined, the…

There is a long history of parabolic monotonicity formulas that developed independently from several different fields and a much more recent elliptic theory. The elliptic theory can be localized and there are additional monotone quantities.…

We establish a family of functional inequalities interpolating between the classical logarithmic Sobolev and Poincar\'e inequalities. We prove that they imply the concentration of measure phenomenon intermediate between Gaussian and…

We investigate links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a density with connected support satisfy a…

Let $\Omega$ be a bounded open set in the Poincar\'e hyperbolic disk, $\mathbb{D}$. In this article, we consider the hyperbolic logarithmic potential operator $\mathcal{L}_h : L^2(\Omega) \to L^2(\Omega)$, defined by \begin{equation*}…

Reverse H\"{o}lder inequalities for a class of functions on a probability space constitute an important tool in Analysis in Probability. After revisiting how a (modified) log-Sobolev inequality can be used to derive reverse H\"{o}lder…

We exhibit several bounds for operator norms of the sum of $\epsilon$-free semicircular random variables introduced in the paper of Speicher and Wysocza\'{n}ski. In particular, using the first and second largest eigenvalues of the adjacency…

In the context of Sobolev spaces with variable exponents, Poincar\'e--Wirtinger inequalities are possible as soon as Luxemburg norms are considered. On the other hand, modular versions of the inequalities in the expected form…

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into $L^q$ spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also…

We investigate the interplay between monomial first integrals, polynomial invariants of certain group action, and the Poincar\'{e}-Dulac normal forms for autonomous systems of ODEs with diagonal matrix of the linear part. Using tools from…

The central aim of this paper is to study (regional) fractional Poincar\'e type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results are established depending on various…

Let $A$ be a rational function of one complex variable, and $z_0$ its repelling fixed point with the multiplier $\lambda.$ Then a Poincar\'e function associated with $z_0$ is a function $\mathcal{P}_{A,z_0,\lambda}$ meromorphic on $\mathbb…

We consider several local versions of the doubling condition and Poincar\'e inequalities on metric spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding…