Hua-harmonic functions on symmetric type two Siegel domains

Dariusz Buraczewski; Ewa Damek

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2002)

  • Volume: 13, Issue: 3-4, page 199-207
  • ISSN: 1120-6330

Abstract

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We study a natural system of second order differential operators on a symmetric Siegel domain D that is invariant under the action of biholomorphic transformations. If D is of type two, the space of real valued solutions coincides with pluriharmonic functions. We show the main idea of the proof and give a survey of previous results.

How to cite

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Buraczewski, Dariusz, and Damek, Ewa. "Hua-harmonic functions on symmetric type two Siegel domains." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13.3-4 (2002): 199-207. <http://eudml.org/doc/252377>.

@article{Buraczewski2002,
abstract = {We study a natural system of second order differential operators on a symmetric Siegel domain $\mathcal\{D\}$ that is invariant under the action of biholomorphic transformations. If $\mathcal\{D\}$ is of type two, the space of real valued solutions coincides with pluriharmonic functions. We show the main idea of the proof and give a survey of previous results.},
author = {Buraczewski, Dariusz, Damek, Ewa},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Symmetric Siegel domain; Pluriharmonic function; Invariant system of differential opera- tors; symmetric Siegel domain; pluriharmonic function; invariant system of differential operators},
language = {eng},
month = {12},
number = {3-4},
pages = {199-207},
publisher = {Accademia Nazionale dei Lincei},
title = {Hua-harmonic functions on symmetric type two Siegel domains},
url = {http://eudml.org/doc/252377},
volume = {13},
year = {2002},
}

TY - JOUR
AU - Buraczewski, Dariusz
AU - Damek, Ewa
TI - Hua-harmonic functions on symmetric type two Siegel domains
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2002/12//
PB - Accademia Nazionale dei Lincei
VL - 13
IS - 3-4
SP - 199
EP - 207
AB - We study a natural system of second order differential operators on a symmetric Siegel domain $\mathcal{D}$ that is invariant under the action of biholomorphic transformations. If $\mathcal{D}$ is of type two, the space of real valued solutions coincides with pluriharmonic functions. We show the main idea of the proof and give a survey of previous results.
LA - eng
KW - Symmetric Siegel domain; Pluriharmonic function; Invariant system of differential opera- tors; symmetric Siegel domain; pluriharmonic function; invariant system of differential operators
UR - http://eudml.org/doc/252377
ER -

References

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  1. Berline, N. - Vergne, M., Equations de Hua et noyau de Poisson. Lecture Notes in Mathematics, vol. 880, Springer-Verlag, 1981, 1-51. Zbl0521.32024MR644825
  2. Bonami, A. - Buraczewski, D. - Damek, E. - Hulanicki, A. - Penney, R. - Trojan, B., Hua system and pluriharmonicity for symmetric irreducible Siegel domains of type II. Journal of Functional Analysis, 188, n. 1, 2002, 38-74. Zbl0999.31005MR1878631DOI10.1006/jfan.2001.3823
  3. Buraczewski, D., Pluriharmonicity of Hua harmonic functions on symmetric irreducible Siegel domains of type II. Preprint. 
  4. Buraczewski, D. - Damek, E. - Hulanicki, A., Bounded plurihramonic functions on symmetric irreducible Siegel domains. Mathematische Zeitschrift, to appear. Zbl1008.32013MR1906712DOI10.1007/s002090100367
  5. Damek, E. - Hulanicki, A., Pluriharmonic functions on symmetric irreducible Siegel domains. Studia Math., 139, n. 2, 2000, 104-140. Zbl0999.32012MR1762449
  6. Damek, E. - Hulanicki, A. - Müller, D. - Peloso, M., Pluriharmonic H 2 functions on symmetric irreducible Siegel domains. Geom. and Funct. Analysis, 10, 2000, 1090-1117. Zbl0969.31007MR1792830DOI10.1007/PL00001648
  7. Harish-Chandra, , Discrete series for semisimple Lie groups II. Acta Math., 116, 1960, 1-111. Zbl0199.20102
  8. Helgason, S., Geometric Analysis on Symmetric Spaces. Amer. Math. Soc., Providence1994. Zbl1157.43003MR1280714
  9. Hua, L.K., Harmonic Analysis of Functions of Several Complex Variables in Classical Domains. Translations of Math. Monographs, vol. 6, Amer. Math. Soc., Providence1963. Zbl0112.07402MR171936
  10. Johnson, K., Remarks on the theorem of Korányi and Malliavin on the Siegel upper half-plane of rank two. Proc. Amer. Math. Soc., 67, 1977, 351-356. Zbl0395.22012MR476918
  11. Johnson, K., Differential equations and the Bergman-Shilov boundary on the Siegel upper half-plane. Arkiv for Matematik, 16, 1978, 95-108. Zbl0395.22013MR499140DOI10.1007/BF02385985
  12. Johnson, K.D. - Korányi, A., The Hua operators on bounded symmetric domains of tube type. Ann. of Math., (2) 111, n. 3, 1980, 589-608. Zbl0468.32007MR577139DOI10.2307/1971111
  13. Korányi, A. - Malliavin, P., Poisson formula and compound diffusion associated to an overdetermined elliptic system on the Siegel halfplane of rank two. Acta Math., 134, 1975, 185-209. Zbl0318.60066MR410278
  14. Lassalle, M., Les équations de Hua d’un domaine borné symétrique du type tube. Invent. Math., 77, 1984, 129-161. Zbl0582.32042MR751135DOI10.1007/BF01389139
  15. Pjatecki-Shapiro, I.I., Geometry and classification of homogeneous bounded domains in C . Uspehi Math. Nauk, 2, 1965, 3-51; Russian Math. Surv., 20, 1966, 1-48. Zbl0142.05002

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