Hardy and Poincar\'e inequalities in fractional Orlicz-Sobolev spaces

TL;DR

This paper establishes sufficient and necessary conditions for Hardy and Poincaré inequalities in fractional Orlicz-Sobolev spaces within various Lipschitz domains, advancing the understanding of these inequalities in complex geometric settings.

Contribution

It provides new sufficient and necessary conditions for Hardy and Poincaré inequalities in fractional Orlicz-Sobolev spaces on Lipschitz domains, extending previous results.

Findings

Sufficient conditions for boundary Hardy inequalities in Lipschitz domains.

Necessary conditions for fractional Orlicz Hardy and Poincaré inequalities.

Various conditions ensuring inequalities hold in different domain types.

Abstract

We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincar\'e inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincar\'e inequalities are also given for bounded Lipschitz domains. Various sufficient conditions on open sets are provided for fractional Orlicz Poincar\'e inequality and regional fractional Orlicz Poincar\'e inequality to hold.