Projections, the weighted Bergman spaces, and the Bloch space

Abstract:

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 ${\mathbb {C}^n}(n \geq 1)$. The corresponding result for the weighted Bergman spaces $A_\alpha ^p$ is obtained. As applications a solution of Gleason’s problem at the origin for $A_\alpha ^p$ and a characterization of $A_\alpha ^p$ in terms of partial derivatives are indicated without proof. Also the natural limiting case is found: ${P_S}{L^\infty } = \mathfrak {B}$, the Bloch space, and ${P_S}{C_0} = {\mathfrak {B}_0}$, the little Bloch space. Moreover, simple bounded linear operators ${L_S}:\mathfrak {B} \to {L^\infty }$, with ${L_S}({\mathfrak {B}_0}) \subset {C_0}$, are found so that ${P_S} \circ {L_S}$ is the identity on $\mathfrak {B}$. As an application the dualities $\mathfrak {B} = {(A_\alpha ^1)^ * }$ and $\mathfrak {B}_0^ * = A_\alpha ^1$ are established under each of pairings suggested by projections ${P_S}$.