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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Projections, the weighted Bergman spaces, and the Bloch space
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by Boo Rim Choe PDF
Proc. Amer. Math. Soc. 108 (1990), 127-136 Request permission


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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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 108 (1990), 127-136
  • MSC: Primary 32A25; Secondary 32A40, 46E99, 47B38
  • DOI:
  • MathSciNet review: 991692