Hardy inequalities on metric measure spaces, II: The case $p>q$

TL;DR

This paper extends the characterization of weights for two-weight Hardy inequalities on metric measure spaces to the case where p > q, providing new inequalities and examples on various geometric spaces without requiring doubling conditions.

Contribution

It offers a novel analysis of Hardy inequalities for p > q on metric measure spaces with polar decompositions, expanding previous results to a broader parameter range.

Findings

New weighted Hardy inequalities on ^n, homogeneous groups, hyperbolic spaces, and Cartan-Hadamard manifolds.

No doubling condition required for the analysis.

Extension of previous work to the case p > q, with integral form inequalities.

Abstract

In this note we continue giving the characterisation of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. This is a continuation of our paper [M. Ruzhansky and D. Verma. Hardy inequalities on metric measure spaces, Proc. R. Soc. A., 475(2223):20180310, 2018] where we treated the case $p\leq q$. Here the remaining range $p>q$ is considered, namely, $0<q<p$, $1<p<\infty.$ We give examples obtaining new weighted Hardy inequalities on $\mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds. We note that doubling conditions are not required for our analysis.