Harmonic Bergman Functions on Half-Spaces

Authors:
Wade C. Ramey and HeungSu Yi

Journal:
Trans. Amer. Math. Soc. **348** (1996), 633-660

MSC (1991):
Primary 31B05; Secondary 31B10, 30D55, 30D45

DOI:
https://doi.org/10.1090/S0002-9947-96-01383-9

MathSciNet review:
1303125

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Abstract | References | Similar Articles | Additional Information

Abstract: We study harmonic Bergman functions on the upper half-space of . Among our main results are: The Bergman projection is bounded for the range ; certain nonorthogonal projections are bounded for the range ; the dual space of the Bergman -space is the harmonic Bloch space modulo constants; harmonic conjugation is bounded on the Bergman spaces for the range ; the Bergman norm is equivalent to a ``normal derivative norm'' as well as to a ``tangential derivative norm''.

**1.**L. Ahlfors,*Some remarks on Teichmüller's space of Riemann surfaces*, Ann. of Math.**74**(1961), 171--191. MR**34:4480****2.**H. Ajmi and W. Ramey,*Harmonic Bloch functions on the upper half space*(to appear).**3.**S. Axler,*Bergman spaces and their operators*, Surveys of Some Recent Results in Operator Theory, Vol. 1, Pitman Research Notes in Math. 171, Pitman, 1988, pp. 1--50. MR**90b:47048****4.**S. Axler, P. Bourdon and W. Ramey,*Harmonic function theory*, Springer-Verlag, New York, 1992. MR**93f:31001****5.**C. Fefferman and E. Stein,*-spaces of several variables*, Acta Math.**129**(1972), 137--193. MR**56:6263****6.**F. Forelli and W. Rudin,*Projections on spaces of holomorphic functions in balls*, Indiana Univ. Math. J.**24**(1974), 593--602. MR**50:10332****7.**G. H. Hardy and J. E. Littlewood,*Some properties of conjugate functions*, J. Reine Angew. Math.**167**(1931), 405--423**8.**H. S. Shapiro,*Global geometric aspects of Cauchy's problem for the Laplace operator*, research report TRITA-MAT-1989-37, Royal Inst. Tech., Stockholm.**9.**A. Shields and D. Williams,*Bounded projections, duality, and multipliers in spaces of analytic functions*, Trans. Amer. Math. Soc.**162**(1971), 287--302. MR**44:790****10.**E. Stein,*Singular integrals and differentiability properties of functions*, Princeton Univ. Press, Princeton, NJ, 1970. MR**44:7280****11.**E. Stein and G. Weiss,*Fourier analysis on Euclidean spaces*, Princeton Univ. Press, Princeton, NJ, 1971. MR**46:4102****12.**K. Zhu,*Operator theory in function spaces*, Marcel Dekker, New York and Basel, 1990. MR**92c:47031**

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Additional Information

**Wade C. Ramey**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Email:
ramey@math.msu.edu

**HeungSu Yi**

Affiliation:
Global Analysis Research Center, Department of Mathematics, Seoul National University, Seoul, Korea #151-742

Email:
hsyi@math.snu.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-96-01383-9

Keywords:
Bergman kernel,
projection operators,
dual spaces,
harmonic Bloch space

Received by editor(s):
October 13, 1994

Article copyright:
© Copyright 1996
American Mathematical Society