Two-weight norm inequalities for potential type and maximal operators in a metric space

TL;DR

This paper characterizes two-weight norm inequalities for potential and maximal operators in metric spaces of homogeneous type, using Sawyer-type testing conditions and dyadic systems, extending Euclidean results to more general spaces.

Contribution

It provides a new characterization of two-weight inequalities in metric spaces without extra geometric assumptions, utilizing dyadic systems for the first time in this context.

Findings

Characterization of two-weight inequalities via Sawyer-type testing conditions

Extension of Euclidean fractional maximal function results to homogeneous type spaces

Use of finite collections of adjacent dyadic systems in the proof

Abstract

We characterize two-weight norm inequalities for potential type integral operators in terms of Sawyer-type testing conditions. Our result is stated in a space of homogeneous type with no additional geometric assumptions, such as group structure or non-empty annulus property, which appeared in earlier works on the subject. One of the new ingredients in the proof is the use of a finite collection of adjacent dyadic systems recently constructed by the author and T. Hyt\"onen. We further extend the previous Euclidean characterization of two-weight norm inequalities for fractional maximal functions into spaces of homogeneous type.