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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
DOI: https://doi.org/10.1090/S0002-9939-1977-0476918-8
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.


References [Enhancements On Off] (What's this?)

  • [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)
  • [5] A. Koranyi and P. Malliavin, Poisson formula and compound diffusion associated to an overdetermined elliptic system on the Siegel halfplane of rank two, Acta Math. 134 (1975), 185-209. MR 0410278 (53:14028)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0476918-8
Keywords: Poisson kernel, Bergman-Šilov boundary, Furstenberg boundary, symplectic transformations
Article copyright: © Copyright 1977 American Mathematical Society

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