Two weight $L^p$ estimates for paraproducts in non-homogeneous settings

TL;DR

This paper establishes a complete characterization of two weight $L^p$ estimates for paraproducts in non-homogeneous settings, highlighting different testing conditions needed depending on whether $p$ is less than or greater than 2.

Contribution

It provides necessary and sufficient Sawyer-type testing conditions for $L^p$ bounds of paraproducts in non-homogeneous contexts, especially for $p eq 2$, involving novel auxiliary operator conditions.

Findings

For $p eq 2$, the conditions differ: only one testing condition for $p o 2$, both for $p > 2$.

The 'adjoint' testing condition involves an auxiliary operator, not the adjoint of the paraproduct.

The results extend the understanding of two weight inequalities in non-homogeneous analysis.

Abstract

We give a necessary and sufficient condition for the two weight $L^p$-estimates for paraproducts in non-homogeneous settings, $1<p<\infty$. We are mainly interested in the case $p\ne 2$, since the case $p=2$ is a well-known and easy corollary of the Carleson embedding theorem. The necessary and sufficient condition is given in terms of testing conditions of Sawyer type: for $p\le 2$ only one ("direct") testing condition is required, but for $p>2$ both "direct" and "adjoint" testing conditions are needed. An interesting feature is that the "adjoint" testing condition is that it is a testing condition not for the adjoint of the paraproduct, but for some auxiliary operator.