Weighted fractional Poincaré inequalities via isoperimetric inequalities

2023/04/05 by Myyryläinen, Kim, Pérez, Carlos, Weigt, Julian · 1 citation
#42B25 #46E35 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.02681

Abstract

Our main result is a weighted fractional Poincaré-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 Poincaré-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 Poincaré inequalities do not hold for p>1.

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• Degenerate Poincaré-Sobolev inequalities via fractional integration

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