KORÁNYI’S LEMMA FOR HOMOGENEOUS SIEGEL DOMAINS OF TYPE II. APPLICATIONS AND EXTENDED RESULTS

2014 ◽  
Vol 90 (1) ◽  
pp. 77-89 ◽  
Author(s):  
DAVID BÉKOLLÉ ◽  
HIDEYUKI ISHI ◽  
CYRILLE NANA
Keyword(s):  
Functional Analysis ◽  
Bergman Kernel ◽  
Atomic Decomposition ◽  
Bergman Spaces ◽  
Decomposition Theorem ◽  
Interpolation Theorem ◽  
Type Ii ◽  
Siegel Domain ◽  
Homogeneous Case ◽  
Siegel Domains

AbstractWe show that the modulus of the Bergman kernel $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}B(z, \zeta )$ of a general homogeneous Siegel domain of type II is ‘almost constant’ uniformly with respect to $z$ when $\zeta $ varies inside a Bergman ball. The control is expressed in terms of the Bergman distance. This result was proved by A. Korányi for symmetric Siegel domains of type II. Subsequently, R. R. Coifman and R. Rochberg used it to establish an atomic decomposition theorem and an interpolation theorem by functions in Bergman spaces $A^p$ on these domains. The atomic decomposition theorem and the interpolation theorem are extended here to the general homogeneous case using the same tools. We further extend the range of exponents $p$ via functional analysis using recent estimates.

scholarly journals On classification of quasi-symmetric domains

1976 ◽  
Vol 62 ◽  
pp. 1-12 ◽  
Author(s):  
I. Satake
Keyword(s):  
Bounded Domain ◽  
Siegel Domain ◽  
The Third ◽  
Siegel Domains

The notion of “Siegel domains” was introduced by [8]. It was then shown that every homogeneous bounded domain is holomorphically equivalent to a Siegel domain (of the second kind) determined uniquely up to an affine isomorphism ([15], cf. also [2], [4], [9b]). In a recent note [10b], I have shown that among (homogeneous) Siegel domains the symmetric domains can be characterized by three conditions (i), (ii), (iii) on the data (U, V, Ω, F) defining the Siegel domain (see Theorem in § 2 of this paper). The class of homogeneous Siegel domains satisfying partial conditions (i), (ii), which we propose to call “quasi-symmetric”, seems to be of some interest, since for instance the fibers appearing in the expressions of symmetric domains as Siegel domains of the third kind fall in this class ([10b], [16]).

scholarly journals On Extremal Problems in Certain New Bergman Type Spaces in Some Bounded Domains inCn

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Romi F. Shamoyan ◽  
Olivera Mihić
Keyword(s):  
Unit Disk ◽  
Bergman Kernel ◽  
Extremal Problems ◽  
Bergman Projection ◽  
Bounded Domains ◽  
Bounded Symmetric Domains ◽  
Siegel Domains ◽  
Type Spaces ◽  
Tube Type ◽  
Homogeneous Domains

Based on recent results on boundedness of Bergman projection with positive Bergman kernel in analytic spaces in various types of domains inCn, we extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains, and minimal bounded homogeneous domains.

ON THE AUTOMORPHISMS AND EQUIVALENCES OF GENERALIZED SIEGEL DOMAINS

1992 ◽  
pp. 495-518 ◽  
Author(s):  
WILHELM KAUP ◽  
YOZO MATSUSHIMA ◽  
TAKUSHIKO OCHIAI
Keyword(s):  
Siegel Domains
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