Two-weight norm inequalities for the Local Maximal Function

TL;DR

This paper characterizes the conditions on weight pairs for the boundedness of a local maximal function within a subset of Euclidean space, extending understanding of weighted inequalities in harmonic analysis.

Contribution

It provides a new characterization of weight pairs ensuring boundedness of a local maximal function on certain weighted Lebesgue spaces.

Findings

Identifies necessary and sufficient conditions for weight pairs (u,v).

Extends classical maximal function inequalities to local settings.

Offers a framework for analyzing weighted inequalities in restricted domains.

Abstract

For a local maximal function defined on a certain family of cubes lying ``well inside'' of $\Omega$, a proper open subset of $\mathbb R ^n$, we characterize the couple of weights $(u,v)$ for which it is bounded from $L^p(v)$ on $L^q(u)$.