Characterization of fractional Sobolev--Poincar\'e and (localized) Hardy inequalities

TL;DR

This paper advances the understanding of fractional Sobolev--Poincaré and Hardy inequalities by establishing capacitary versions, characterizing boundary inequalities via domain fatness, and exploring existence and characterization results in supercritical cases.

Contribution

It introduces capacitary versions of fractional inequalities, characterizes boundary inequalities using fatness conditions, and provides existence and characterization results for fractional Hardy inequalities in supercritical regimes.

Findings

Capacitary versions of fractional Sobolev--Poincaré inequalities are proved.

Localized boundary inequalities are characterized through uniform fatness conditions.

Existence of fractional Hardy inequalities is established in the supercritical case $sp>n$.

Abstract

In this paper, we prove capacitary versions of the fractional Sobolev--Poincar\'e inequalities. We characterize localized variant of the boundary fractional Sobolev--Poincar\'e inequalities through uniform fatness condition of the domain in $\mathbb{R}^n$. Existence type results on the fractional Hardy inequality are established in the supercritical case $sp>n$ for $s\in(0,1)$, $p>1$. Characterization of the fractional Hardy inequality through weak supersolution of the associate problem is also addressed.