Weighted Bloch, Lipschitz, Zygmund, Bers, and Growth Spaces of the Ball: Bergman Projections and Characterizations

Abstract

We determine precise conditions for the boundedness of Bergman projections from Lebesgue classes onto the spaces in the title, which are members of the same one-parameter family of spaces. The projections provide integral representations for the functions in the spaces. We obtain many properties of the spaces as straightforward corollaries of the projections, integral representations, and isometries among the spaces. We solve the Gleason problem and an extremal problem for point evaluations in each space. We establish maximality of these spaces among those that exhibit Mobius-type invariances and possess decent functionals. We find new Hermitian non-Kahlerian metrics that characterize half of these spaces by Lipschitz-type inequalities.