The Bergman spaces, the Bloch space, and Gleason's problem

Author:
Ke He Zhu

Journal:
Trans. Amer. Math. Soc. **309** (1988), 253-268

MSC:
Primary 46E15; Secondary 32A35, 32H10, 46J15

MathSciNet review:
931533

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Abstract: Suppose is a holomorphic function on the open unit ball of . For and an integer, we show that is in (with the volume measure) iff all the functions are in . We also prove that is in the Bloch space of iff all the functions are bounded on . The corresponding result for the little Bloch space of is established as well. We will solve Gleason's problem for the Bergman spaces and the Bloch space of before proving the results stated above. The approach here is functional analytic. We make extensive use of the reproducing kernels of . The corresponding results for the polydisc in are indicated without detailed proof.

**[1]**Sheldon Axler,*Bergman spaces and their operators*, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 1–50. MR**958569****[2]**R. R. Coifman and R. Rochberg,*Representation theorems for holomorphic and harmonic functions*, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 459–460. MR**545288****[3]**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****[4]**Frank Forelli and Walter Rudin,*Projections on spaces of holomorphic functions in balls*, Indiana Univ. Math. J.**24**(1974/75), 593–602. MR**0357866****[5]**Walter Rudin,*Function theory in the unit ball of 𝐶ⁿ*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594****[6]**A. Zabulionsis,*Differential operators in spaces of analytic function*, Lithuanian Math. J.**24**(1984), 32-36.**[7]**Ke He Zhu,*Duality and Hankel operators on the Bergman spaces of bounded symmetric domains*, J. Funct. Anal.**81**(1988), no. 2, 260–278. MR**971880**, 10.1016/0022-1236(88)90100-0

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DOI:
https://doi.org/10.1090/S0002-9947-1988-0931533-6

Article copyright:
© Copyright 1988
American Mathematical Society