Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Projections, the weighted Bergman spaces, and the Bloch space

Author: Boo Rim Choe
Journal: Proc. Amer. Math. Soc. 108 (1990), 127-136
MSC: Primary 32A25; Secondary 32A40, 46E99, 47B38
MathSciNet review: 991692
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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}$.

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Keywords: Projections, weighted Bergman spaces, Bloch space
Article copyright: © Copyright 1990 American Mathematical Society