Improved Poincar\'e inequalities in fractional Sobolev spaces

TL;DR

This paper derives enhanced fractional Poincaré inequalities that incorporate boundary distance powers in various domain types, advancing the understanding of fractional Sobolev spaces.

Contribution

It introduces improved fractional Poincaré inequalities with boundary distance powers for specific domain classes, discussing their optimality.

Findings

Enhanced inequalities in John and H"older-$\alpha$ domains

Inclusion of boundary distance powers improves inequality bounds

Discussion on the optimality of the derived inequalities

Abstract

We obtain improved fractional Poincar\'e and Sobolev Poincar\'e inequalities including powers of the distance to the boundary in John, $s$-John domains and H\"older-$\alpha$ domains, and discuss their optimality.