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Remarks on a theorem of Koranyi and Malliavin on the Siegel upper half plane of rank two

Author: Kenneth D. Johnson
Journal: Proc. Amer. Math. Soc. 67 (1977), 351-356
MSC: Primary 22E30; Secondary 32M15
MathSciNet review: 0476918
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Abstract: In [5], A. Koranyi and P. Malliavin showed that bounded functions on the Siegel upper half plane of rank two which satisfied two special elliptic differential equations were characterized by their values on the Bergman-Šilov boundary. In this paper a simple proof of this theorem is given.

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  • [1] H. Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. 77 (1963), 335-386. MR 0146298 (26:3820)
  • [2] L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Transl. Math. Monographs, vol. 6, Amer. Math. Soc., Providence, R. I., 1963. MR 0171936 (30:2162)
  • [3] S. Helgason and A. Koranyi, A Fatou-type theorem for harmonic functions on symmetric spaces, Bull. Amer. Math. Soc. 74 (1968), 258-263. MR 0229179 (37:4753)
  • [4] K. D. Johnson, Differential equations and the Bergman-Šilov boundary on the Siegel upper halfplane, Ark. Mat. (to appear). MR 499140 (80d:32032)
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Keywords: Poisson kernel, Bergman-Šilov boundary, Furstenberg boundary, symplectic transformations
Article copyright: © Copyright 1977 American Mathematical Society

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