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 .