Two weight L^{p} inequalities for fractional vector Riesz transforms and doubling measures

TL;DR

This paper establishes necessary and sufficient conditions for two-weight L^{p} inequalities involving fractional vector Riesz transforms and doubling measures, expanding the understanding of weighted inequalities in harmonic analysis.

Contribution

It proves the equivalence of quadratic triple testing conditions with two-weight inequalities and introduces relaxed testing and Muckenhoupt conditions.

Findings

Quadratic triple testing conditions are necessary and sufficient.

Relaxed conditions also characterize the inequalities.

Results apply to doubling measures and fractional vector Riesz transforms.

Abstract

If T is a fractional vector Riesz transform, 1<p<infinity, and sigma and omega are doubling measures, then the two weight L^{p} norm inequality holds if and only if the quadratic triple testing conditions of Hyt\"onen and Vuorinen hold. We also show that these quadratic triple testing conditions can be relaxed to quadratic local testing conditions, quadratic offset Muckenhoupt conditions, and a quadratic weak boundedness property.