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

摘要

We give a necessary and sufficient condition for the two weight L-p-estimates for paraproducts in non-homogeneous settings, 1 p infinity. We are mainly interested in the case p not equal 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 = 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. (C) 2017 Elsevier Inc. All rights reserved.