Hardy inequalities on metric measure spaces, III: The case $q\leq p<0$ and applications

TL;DR

This paper develops reverse Hardy inequalities with negative exponents on metric measure spaces and applies them to establish new reverse Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups, extending previous results.

Contribution

It introduces reverse Hardy inequalities for the case $q \,\leq\, p < 0$ and applies these to derive novel inequalities on Lie groups, expanding the known parameter ranges.

Findings

Established reverse Hardy inequalities with negative exponents.

Derived new reverse Hardy-Littlewood-Sobolev inequalities.

Extended inequalities to homogeneous Lie groups with arbitrary quasi-norm.

Abstract

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. Also, as for applications we show the reverse Hardy-Littlewood-Sobolev and the Stein-Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result which appears to be new already in the Euclidean space. This work further complements the ranges of $p$ and $q$ (namely, $q\leq p<0$) considered in \cite{RV} and \cite{RV21}, where one treated the cases $1<p\leq q<\infty$ and $p>q$, respectively.