Two weight Lp estimates for paraproducts in non-homogeneous settings

We give a necessary and sufficient condition for the two weight Lp-estimates for paraproducts in non-homogeneous settings, 12 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.