Fractional Poincar\'e and localized Hardy inequalities on metric spaces

TL;DR

This paper establishes new fractional Sobolev-Poincaré, Hardy, and capacitary inequalities in metric spaces with doubling measures, extending previous Euclidean and non-fractional results, with novel pointwise and localized variants.

Contribution

It introduces and proves fractional Hardy inequalities and their localized versions in metric spaces, generalizing classical results and providing new insights even in Euclidean spaces.

Findings

Proved fractional Sobolev-Poincaré inequalities in metric spaces.

Established capacitary fractional Poincaré inequalities.

Derived new pointwise and localized fractional Hardy inequalities.

Abstract

We prove fractional Sobolev-Poincar\'e inequalities, capacitary versions of fractional Poincar\'e inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results generalize and extend earlier work where such inequalities have been considered in the Euclidean spaces or in the non-fractional setting in metric spaces. The results concerning pointwise and localized variants of fractional Hardy inequalities are new even in the Euclidean case.