Reverse integral Hardy inequality on metric measure spaces

TL;DR

This paper establishes a reverse integral Hardy inequality on metric measure spaces, providing necessary and sufficient conditions, and explores its implications on various geometric structures.

Contribution

It introduces a reverse Hardy inequality on metric measure spaces with conditions and applies it to homogeneous groups, hyperbolic spaces, and Cartan-Hadamard manifolds.

Findings

Derived necessary and sufficient conditions for the reverse Hardy inequality.

Extended the inequality to specific geometric contexts like homogeneous groups.

Connected the inequality to the reverse Minkowski inequality.

Abstract

In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski inequality. Also, we present some consequences of the obtained reverse Hardy inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds.