Sharp weighted norm estimates for the positive Bergman operator: the two-parameter case

Abstract

We settle the two-parameter sharp weighted theory for the positive Bergman operator on the upper half-plane: we allow the exponent $α$ that defines the operator to differ from the exponent $γ$ that defines the underlying weighted Lebesgue spaces. In this setting, we prove sharp weak-type and strong-type off-diagonal weighted inequalities, with explicit and best-possible dependence on the Békollé--Bonami characteristic of the weight. The key new tool is an off-diagonal extrapolation theorem, adapted to allow a change in the power of the distance to the boundary along the way.