On how Poincaré inequalities imply weighted ones
type: Item
instance of: scholarly article
title: On how Poincaré inequalities imply weighted ones
zbMATH Open document ID: 1414.26036
author: Martin Rathmair
published in: Monatshefte für Mathematik
publication date: 2019-03-29T00:00:00Z
review text: The present paper is devoted to Poincaré type inequalities that bound the $L_q$-norm of a function $f$ on a domain \(\Omega\) equipped with a non-negative weight $w$ in terms of the $L_p$-norm of its gradient in the form \[ \|f\|_{L^q(\Omega,w)}\le c\|\nabla f\|_{L^p(\Omega,w)} \] where the finite constant $c$ is independent of $f$ and the inequality holds for every $f\in \mathcal{C}$. The space $\mathcal{C}$ considered by the author is a vector space of differentiable (at least in a weak sense) functions $f$ on $\Omega$ satisfying the equality $\int_{\Omega}{fw}=0$ for each $f\in \mathcal{C}$. The main result of this paper is that the Poincaré constant w.r.t. a weight $w$ can be controlled in terms of the Poincaré constants w.r.t. the constant weight 1 on the superlevel sets of $w$. It turns out that it is essentially sufficient that on all superlevel sets of $w$ there hold Poincaré inequalities w.r.t. the constant weight 1 and that the corresponding Poincaré constants satisfy an integrability condition.
reviewed by: Andrey Zahariev
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 35A23
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 7045090
zbMATH Keywords: weighted Poincaré inequality
zbMATH Keywords: Poincaré constant
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: superlevel sets
MaRDI profile type: Publication
full work available at URL: s00605-019-01266-w
OpenAlex ID: W2914124241
cites work: An optimal Poincaré inequality in $L^1$ for convex domains
cites work: Stable phase retrieval in infinite dimensions
cites work: Some Remarks on Rellich's Theorem and the Poincaré Inequality
cites work: A note on the Poincaré inequality for convex domains
cites work: Isoperimetric and analytic inequalities for log-concave probability measures
cites work: An optimal Poincaré-Wirtinger inequality in Gauss space
cites work: Q5614192
cites work: Sharp conditions for 1-dimensional weighted Poincare inequalities
cites work: ESTIMATES OF BEST CONSTANTS FOR WEIGHTED POINCARÉ INEQUALITIES ON CONVEX DOMAINS
cites work: Les espaces du type de Beppo Levi
cites work: Improved Poincaré inequalities with weights
cites work: On weighted Poincaré inequalities
cites work: Weighted Poincaré and Friedrichs Inequalities
cites work: Theory and practice of finite elements.
cites work: Best constants in Poincar\'e inequalities for convex domains
cites work: The local regularity of solutions of degenerate elliptic equations
cites work: Note on Friedrichs' inequality in \(N\)-star-shaped domains
cites work: Stable Gabor Phase Retrieval and Spectral Clustering
cites work: Q3797452
cites work: Weighted Norm Inequalities for the Hardy Maximal Function
cites work: An optimal Poincaré inequality for convex domains
DOI: 10.1007/S00605-019-01266-W
description: scientific article; zbMATH DE number 7045090