Characterization of fractional Sobolev--Poincaré and (localized) Hardy inequalities

Abstract

In this paper, we prove capacitary versions of the fractional Sobolev--Poincaré inequalities. We characterize localized variant of the boundary fractional Sobolev--Poincaré 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.

19 pages, We removed theorem 1.12 from the 1st version of the paper. There were some incorrect results and arguments. We correct them all, especially in the proof of theorem 1.9

References in corpus (3)

• On density of compactly supported smooth functions in fractional Sobolev spaces
• An optimal lower bound in fractional spectral geometry for planar sets with topological constraints
• Fractional Poincaré and localized Hardy inequalities on metric spaces

Cited by in corpus (2)

• Weighted fractional Hardy inequalities with singularity on any flat submanifold
• A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces