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

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.