Weighted fractional Poincar\'e inequalities via isoperimetric inequalities

TL;DR

This paper introduces a new weighted fractional Poincaré-Sobolev inequality that improves existing estimates and extends classical results using fractional isoperimetric inequalities, with implications for weighted analysis.

Contribution

It provides a novel weighted fractional Poincaré-Sobolev inequality based on fractional isoperimetric inequalities, improving classical bounds and extending weighted Poincaré estimates.

Findings

Improved fractional Poincaré-Sobolev inequality over Bourgain-Brezis-Mironescu

Extension of Meyers-Ziemer theorem with fractional methods

Demonstration that weighted Poincaré inequalities do not hold for p>1

Abstract

Our main result is a weighted fractional Poincar\'e-Sobolev inequality improving the celebrated estimate by Bourgain-Brezis-Mironescu. This also yields an improvement of the classical Meyers-Ziemer theorem in several ways. The proof is based on a fractional isoperimetric inequality and is new even in the non-weighted setting. We also extend the celebrated Poincar\'e-Sobolev estimate with $A_p$ weights of Fabes-Kenig-Serapioni by means of a fractional type result in the spirit of Bourgain-Brezis-Mironescu. Examples are given to show that the corresponding $L^p$-versions of weighted Poincar\'e inequalities do not hold for $p>1$.