Projections, the weighted Bergman spaces, and the Bloch space

Abstract

It has been known that there is a family of projections Ps of the Lebesgue spaces onto the Bergman spaces on the unit ball of C(n > 1). The corresponding result for the weighted Bergman spaces Apa is obtained. As applications a solution of Gleason's problem at the origin for Apa and a characterization of Apa in terms of partial derivatives are indicated without proof. Also the natural limiting case is found: PsL°° = S , the Bloch space, and PsCq = *Bo > the I'tt'e Bloch space. Moreover, simple bounded linear operators Ls : 2$ —> L°° , with LsC&q) c Co , are found so that Ps o Ls is the identity on 05 . As an application the dualities 03 = (Axa)* and 23J = Ala are established under each of pairings suggested by projections Ps .