On the BBM-phenomenon in fractional Poincar\'e-Sobolev inequalities with weights

TL;DR

This paper extends fractional Poincaré-Sobolev inequalities with weights, unifying previous results and analyzing the behavior of constants as the fractional parameter approaches one, using harmonic analysis techniques.

Contribution

It provides weighted fractional Poincaré-Sobolev and Hardy inequalities with detailed constant behavior analysis near the classical limit, improving and unifying prior results.

Findings

Weighted fractional inequalities with explicit constant behavior

Unification of Bourgain, Brezis, Mironescu, and Fabes et al. results

Analysis of constants as fractional parameter approaches 1

Abstract

In this paper we unify and improve some of the results of Bourgain, Brezis and Mironescu and the weighted Poincar\'e-Sobolev estimate by Fabes, Kenig and Serapioni. More precisely, we get weighted counterparts of the Poincar\'e-Sobolev type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the fact that we keep track of the behaviour of the constants involved when the fractional parameter approaches to $1$. Our main method is based on techniques coming from harmonic analysis related to the self-improving property of generalized Poincar\'e inequalities.