Projections, the weighted bergman spaces, and the bloch space

It has been known that there is a family of projections P_s of the Lebesgue spaces onto the Bergman spaces on the unit ball of ℂ_n(n ≥ 1). The corresponding result for the weighted Bergman spaces A^pα is obtained. As applications a solution of Gleason’s problem at the origin for Apa and a characterization of A^pα in terms of partial derivatives are indicated without proof. Also the natural limiting case is found: P_sL^∞ = S, the Bloch space, and P_sL^∞ = B the P_sC₀ Bloch space. Moreover, simple bounded linear operators L_s: B →s L^∞ B = (A¹α)* and B₀* = A¹α are established under each of pairings suggested by projections P_s.