Stein’s Theorem in the Upper-Half Plane and Bergman Spaces with Weights

This is a companion paper to our previous one, Avatars of Stein’s Theorem in the complex setting. In this previous paper, we gave a sufficient condition for an integrable function in the upper-half plane to have an integrable Bergman projection. Here we push forward methods and establish in particular a converse statement. This naturally leads us to study a family of weighted Bergman spaces for logarithmic weights \((1+\ln _+(1/\Im m (z))+\ln _+(|z|))^k,\) which have the same kind of behavior, respectively, at the boundary and at infinity. We introduce their duals, which are logarithmic Bloch type spaces and interest ourselves in multipliers, pointwise products, and Hankel operators. In particular we characterize the symbols of bounded Hankel operators within these weighted Bergman spaces.