Hardy inequalities on metric measure spaces

TL;DR

This paper characterizes weights for Hardy inequalities on general metric measure spaces with polar decompositions, providing new integral inequalities applicable to various geometric contexts like Euclidean spaces, groups, and manifolds.

Contribution

It introduces several characterizations of weights for Hardy inequalities on metric measure spaces without differentiable structures, extending classical results to broader geometric settings.

Findings

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

Characterizations of weights for two-weight Hardy inequalities in metric measure spaces.

Applications to diverse geometric contexts without relying on differentiability.

Abstract

In this note we give several characterisations 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. We give examples obtaining new weighted Hardy inequalities on $\mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.