Let be an irreducible Hermitian symmetric space of noncompact type. We study a - invariant system of differential operators on called the Hua system. It was proved by K. Johnson and A. Korányi that if is a Hermitian symmetric space of tube type, then the space of Poisson-Szegö integrals is precisely the space of zeros of the Hua system. N. Berline and M. Vergne raised the question about the nature of the common solutions of the Hua system for Hermitian symmetric spaces of nontube type. In this paper we show that these are exactly the pluriharmonic functions.
Soit un espace hermitien symétrique non compact irréductible. On étudie un système invariant d’opérateurs differentiels sur . Selon un théorème de K. Johnson et A. Korányi, une fonction sur un espace hermitien symétrique de type tube est annulée par le système de Hua si et seulement si elle est l’intégrale de Poisson-Szegö d’une hyperfonction. N. Berline et M. Vergne ont posé la question de caractériser les fonctions - harmoniques sur les espaces hermitiens de type II. Ici on montre que ce sont les fonctions pluriharmoniques.
Keywords: pluriharmonic functions, Hua system, Hermitian symmetric spaces, Siegel domains
Mot clés : fonctions pluriharmoniques, système de Hua, espaces hermitiens, domaines symétriques
@article{AIF_2004__54_1_81_0, author = {Buraczewski, Dariusz}, title = {The {Hua} system on irreducible {Hermitian} symmetric spaces of nontube type}, journal = {Annales de l'Institut Fourier}, pages = {81--127}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {54}, number = {1}, year = {2004}, doi = {10.5802/aif.2011}, mrnumber = {2069122}, zbl = {1065.32017}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2011/} }
TY - JOUR AU - Buraczewski, Dariusz TI - The Hua system on irreducible Hermitian symmetric spaces of nontube type JO - Annales de l'Institut Fourier PY - 2004 SP - 81 EP - 127 VL - 54 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2011/ DO - 10.5802/aif.2011 LA - en ID - AIF_2004__54_1_81_0 ER -
%0 Journal Article %A Buraczewski, Dariusz %T The Hua system on irreducible Hermitian symmetric spaces of nontube type %J Annales de l'Institut Fourier %D 2004 %P 81-127 %V 54 %N 1 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2011/ %R 10.5802/aif.2011 %G en %F AIF_2004__54_1_81_0
Buraczewski, Dariusz. The Hua system on irreducible Hermitian symmetric spaces of nontube type. Annales de l'Institut Fourier, Volume 54 (2004) no. 1, pp. 81-127. doi : 10.5802/aif.2011. http://archive.numdam.org/articles/10.5802/aif.2011/
[BBDHPT] Hua system and pluriharmonicity for symmetric irreducible Siegel domains of type II, Journal of Functional Analysis, Volume 188 (2002), pp. 38-74 | MR | Zbl
[BDH] Bounded pluriharmonic functions on symmetric irreducible Siegel domains, Mathematische Zeitschrift, Volume 240 (2002), pp. 169-195 | MR | Zbl
[BV] Equations de Hua et noyau de Poisson (Lecture Notes in Math.), Volume 880 (1981), pp. 1-51 | MR | Zbl
[DH] Boundaries for left-invariant subelliptic operators on semidirect products of nilpotent and abelian groups, J. Reine Angew. Math, Volume 411 (1990), pp. 1-38 | MR | Zbl
[DHMP] \newblock Pluriharmonic functions on symmetric irreducible Siegel domains, Geom. and Funct. Anal, Volume 10 (2000), pp. 1090-1117 | MR | Zbl
[DHP] Hua operators on bounded homogeneous domains in and alternative reproducing kernels for holomorphic functions, Journal of Functional Analysis, Volume 151 (1997) no. 1, pp. 77-120 | MR | Zbl
[FK] Analysis On Symmetric Cones, Clarendon Press, Oxford, 1994 | MR | Zbl
[H1] Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1962 | MR | Zbl
[H2] Groups and Geometric Analysis, Academic Press, Orlando, 1984 | MR | Zbl
[H3] Geometric Analysis on Symmetric Spaces, American Mathematical Society, Providence, 1994 | MR | Zbl
[HC] Discrete series for semisimple Lie groups II, Acta Math, Volume 116 (1966), pp. 1-111 | MR | Zbl
[Hua] Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Science Press, Peking (1958) (Math. Monograph) (1963) | MR | Zbl
[J1] Remarks on the theorem of Korányi and Malliavin on the Siegel upper half-plane of rank two, Proc. Amer. Math. Soc, Volume 67 (1977), pp. 351-356 | MR | Zbl
[J2] Differential equations and the Bergman-Shilov boundary on the Siegel upper half-plane, Arkiv for Matematik, Volume 16 (1978), pp. 95-108 | MR | Zbl
[JK] The Hua operators on bounded symmetric domains of tube type, Annals of Math, Volume 111 (1980) no. 2, pp. 589-608 | MR | Zbl
[K] Analysis and Geometry on Complex Homogeneous Domains, chapter Function Spaces on Bounded Symmetric Domains (2000), pp. 183-281 | MR | Zbl
[KM] Poisson formula and compound diffusion associated to an overdetermined elliptic system on the Siegel halfplane of rank two, Acta Math, Volume 134 (1975), pp. 185-209 | MR | Zbl
[Kn] Lie Groups Beyond an Introduction, Birkhäuser, Boston--Basel--Berlin, 1996 | MR | Zbl
[KV] Rational inner functions on bounded symmetric domains, Trans. A. M. S, Volume 254 (1979), pp. 179-193 | MR | Zbl
[KW] Realization of Hermitian symmetric spaces as generalized half-planes, Annals of Math, Volume 81 (1965) no. 2, pp. 265-288 | MR | Zbl
[L] Les équations de Hua d'un domaine borné symétrique de tube type, Invent. Math, Volume 77 (1984), pp. 129-161 | MR | Zbl
[R] Fonctions harmoniques sur les groupes localement compact à base dénombrable, Bull. Soc. Math. France, Mémoire, Volume 54 (1977), pp. 5-118 | Numdam | MR | Zbl
[S] Algebraic structures of symmetric domains, Iwanami-Shoten and Princeton Univ. Press, 1980 | MR | Zbl
[T] Harmonic Analysis on the Heisenberg Group, Birkhäuser, Boston--Basel--Berlin, 1998 | MR | Zbl
Cited by Sources: