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

Published online by Cambridge University Press:  13 May 2014

DAVID BÉKOLLÉ*
Affiliation:
University of Ngaoundéré, Faculty of Science, Department of Mathematics and Computer Science, PO Box 454, Ngaoundéré, Cameroon email bekolle@yahoo.fr
HIDEYUKI ISHI
Affiliation:
Graduate School of Mathematics, Nagoya University, Furo-Cho, Chikusa-Ku, Nagoya 464-8602, Japan email hideyuki@math.nagoya-u.ac.jp
CYRILLE NANA
Affiliation:
University of Buea, Faculty of Science, Department of Mathematics, PO Box 63, Buea, Cameroon email nana.cyrille@ubuea.cm
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Abstract

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We 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.

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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