Hardy's inequalities for monotone functions on partially ordered measure spaces

TL;DR

This paper extends the characterization of weighted Hardy's inequalities for monotone functions from the classical one-dimensional case to higher dimensions and more general partially ordered measure spaces, broadening the theoretical understanding.

Contribution

It generalizes Hardy's inequalities to n-dimensional partially ordered measure spaces, including the case p>1, which was previously known only for p=1.

Findings

Characterization of weighted Hardy's inequalities in higher dimensions

Extension of classical theory of B_p weights to n>1

Main theorem proved in general partially ordered measure spaces

Abstract

We characterize the weighted Hardy's inequalities for monotone functions in ${\mathbb R^n_+}.$ In dimension $n=1$, this recovers the classical theory of $B_p$ weights. For $n>1$, the result was only known for the case $p=1$. In fact, our main theorem is proved in the more general setting of partially ordered measure spaces.