On fractional inequalities on metric measure spaces with polar decomposition

TL;DR

This paper establishes various fractional inequalities, including Hardy, Hardy-Sobolev, and Nash types, on polarizable metric measure spaces, with applications to homogeneous groups and the Heisenberg group.

Contribution

It introduces fractional inequalities on metric measure spaces with polar decomposition, extending classical results to more general geometric contexts.

Findings

Proved fractional Hardy inequality on polarizable metric measure spaces

Established fractional Hardy-Sobolev and Nash inequalities in this setting

Applied results to homogeneous groups and the Heisenberg group

Abstract

In this paper, we prove the fractional Hardy inequality on polarisable metric measure spaces. The integral Hardy inequality for $1<p\leq q<\infty$ is playing a key role in the proof. Moreover, we also prove the fractional Hardy-Sobolev type inequality on metric measure spaces. In addition, logarithmic Hardy-Sobolev and fractional Nash type inequalities on metric measure spaces are presented. In addition, we present applications on homogeneous groups and on the Heisenberg group.