A Two-Weight Inequality for Essentially Well Localized Operators with General Measures

TL;DR

This paper introduces a new formulation and proof for characterizing the boundedness of well localized operators between weighted L^2 spaces with general measures, advancing the understanding of two-weight inequalities.

Contribution

It provides a novel formulation of well localized operators and a new proof for the necessary and sufficient conditions for their boundedness with general Radon measures.

Findings

Established a new formulation of well localized operators.

Derived necessary and sufficient conditions for boundedness between weighted spaces.

Extended the theory to general Radon measures.

Abstract

We develop a new formulation of well localized operators as well as a new proof for the necessary and sufficient conditions to characterize their boundedness between $L^2(\mathbb{R}^n,u)$ and $L^2(\mathbb{R}^n,v)$ for general Radon measures $u$ and $v$.