Two Weight Inequalities for Discrete Positive Operators

TL;DR

This paper characterizes two weight inequalities for positive dyadic operators, extending classical results and employing an adapted Sawyer's method to analyze weak and strong type inequalities across various (p,q) mappings.

Contribution

It provides a unified characterization of two weight inequalities for positive dyadic operators, extending previous classical and bilinear results with a novel proof approach.

Findings

Characterization of two weight inequalities for positive dyadic operators

Extension of Sawyer's argument to new settings

Connections to classical fractional integral and bilinear embedding results

Abstract

We characterize two weight inequalities for general positive dyadic operators. We consider both weak and strong type inequalities, and general (p,q) mapping properties. Special cases include Sawyers Fractional Integral operator results from 1988, and the bilinear embedding inequality of Nazarov-Treil-Volberg from 1999. The method of proof is an extension of Sawyer's argument.