\(L^1\rightarrow L^q\) Poincaré inequalities for \(0<q<1\) imply representation formulas
type: Item
instance of: scholarly article
title: \(L^1\rightarrow L^q\) Poincaré inequalities for \(0<q<1\) imply representation formulas
zbMATH Open document ID: 1012.46037
author: Guozhen Lu
author: Carlos Pérez
publication date: 2002-08-06T00:00:00Z
review text: The authors shows that an \(L^1\) to \(L^q\) Poincaré inequality for some \(0<q<1\), i.e., \[ \left(\not \int_B|f-f_B |^q d\nu \right)^{1/q}\leq Cr(B) \not \int_B gd\mu, \] for all metrie balls \(B \subset B_0\), will suffice to imply the representation formula. As an immediate corollary, the authors can show that the weak-type condition \[ \sup_{ \lambda>0} {\lambda\nu \Bigl(\biggl\{x\in B:\bigl|f(x)-f_B\bigr|>\lambda \biggr\} \Bigr) \over\nu(B)}\leq Cr(B) \not \int_B gd\mu, \] also implies the same formula. Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved.
reviewed by: Stanisław Wedrychowicz
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 1757441
zbMATH Keywords: representation formula
zbMATH Keywords: Poincaré inequality
MaRDI profile type: Publication
full work available at URL: s101140100154
OpenAlex ID: W2065963868
cites work: Q4882798
cites work: Some remarks about Poincaré type inequalities and representation formulas in metric spaces of homogeneous type
cites work: An Optimal Representation Formula for Carnot-Carathéodory Vector Fields
cites work: High order representation formulas and embedding theorems on stratified groups and generalizations
cites work: Q3700300
cites work: Definitions of Sobolev classes on metric spaces
cites work: Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces
cites work: Q3993586
cites work: Q4847438
cites work: Sobolev inequalities on homogeneous spaces
cites work: Analysis on geodesic balls of sub-elliptic operators
cites work: Isoperimetric and Sobolev inequalities for Carnot-Carath�odory spaces and the existence of minimal surfaces
cites work: Sobolev inequalities in disguise
cites work: Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type
cites work: The Poincaré inequality for vector fields satisfying Hörmander's condition
cites work: The sharp Poincaré inequality for free vector fields: An endpoint result
cites work: Representation formulas and weighted Poincaré inequalities for Hörmander vector fields
cites work: Weighted Sobolev-Poincare Inequalities and Pointwise Estimates for a Class of Degenerate Elliptic Equations
cites work: Q3345173
cites work: Q4735511
cites work: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications
cites work: Weighted sobolev-poincaré inequalities for grushin type operators
cites work: Subelliptic mollifiers and a basic pointwise estimate of Poincaré type
cites work: Q3811180
cites work: Weighted Sobolev Inequalities on Domains Satisfying the Chain Condition
cites work: Hardy-Littlewood inequality for quasiregular mappings in certain domains in R^n
cites work: On uniform properties of doubling measures
cites work: Q4870771
cites work: Local and global interpolation inequalities on the Folland-Stein Sobolev spaces and polynomials on stratified groups
cites work: Polynomials, higher order Sobolev extension theorems and interpolation inequalities on weighted Folland-Stein spaces on stratified groups
cites work: Q4494980
cites work: Q4494979
DOI: 10.1007/S101140100154
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1757441