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

Abstract

We allow the exponent $\alpha$ that defines the positive Bergman operator on the upper half-plane to differ from the exponent $\gamma$ that defines the underlying weighted Lebesgue spaces. In this setting, we prove a sharp weak-type off-diagonal weighted inequality, with explicit and best-possible dependence on the Békollé--Bonami characteristic of the weight. We then deduce strong-type off-diagonal inequalities for the same operator. 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.

11 pages