Hardy inequalities on metric measure spaces, IV: The case $p=1$

TL;DR

This paper extends Hardy inequalities to the case p=1 on metric measure spaces with polar decompositions, providing new results and best constants, especially on Lie groups, hyperbolic spaces, and Cartan-Hadamard manifolds.

Contribution

It develops the Hardy inequalities for p=1, which were not covered by previous work, and determines the best constants in these inequalities.

Findings

Established new weighted Hardy inequalities for p=1.

Provided explicit best constants in the inequalities.

Presented examples on Lie groups, hyperbolic spaces, and Cartan-Hadamard manifolds.

Abstract

In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case $p=1$ and $1 \leq q <\infty.$ This result complements the Hardy inequalities obtained in \cite{RV} in the case $1< p\le q<\infty.$ The case $p=1$ requires a different argument and does not follow as the limit of known inequalities for $p>1.$ As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan-Hadamard manifolds for the case $p=1$ and $1\le q<\infty.$