Iterates and the boundary behavior of the Berezin transform

Arazy, Jonathan, and Engliš, Miroslav. "Iterates and the boundary behavior of the Berezin transform." Annales de l’institut Fourier 51.4 (2001): 1101-1133. <http://eudml.org/doc/115936>.

@article{Arazy2001,
abstract = {Let $\mu $ be a measure on a domain $\Omega $ in $\{\mathbb \{C\}\}^n$ such that the Bergman space of holomorphic functions in $L^2(\Omega ,\mu )$ possesses a reproducing kernel $K(x,y)$ and $K(x,x)&gt;0$$\forall x\in \Omega $. The Berezin transform associated to $\mu $ is the integral operator\[ Bf(y) = K(y,y)^\{-1\} \int \_\Omega f(x)\vert K(x,y)\vert ^2 \,d\mu (x). \]The number $Bf(y)$ can be interpreted as a certain mean value of $f$ around $y$, and functions satisfying $Bf=f$ as functions having a certain mean-value property. In this paper we investigate the boundary behavior of $Bf$, the existence of functions $f$ satisfying $Bf=f$ and having prescribed boundary values, and the convergence of the iterates $B^k f$, $k\rightarrow \infty $. The best results are obtained for smoothly bounded strictly pseudoconvex domains $\Omega $ with any measure $\mu $ as above, and for bounded symmetric domains $\Omega $ and $\mu $ one of the standard rotation-invariant measures on them. We also carry out similar investigation for convolution operators $B_\mu f=f*\mu $ on a bounded symmetric domain $\Omega =G/K$ with a $K$-invariant absolutely continuous probability measure $\mu $, and study the behavior of the geodesic symmetries $\phi _a$ of $\Omega $ as $a$ tends to the boundary.},
affiliation = {University of Haifa, Department of Mathematics, Haifa 31905 (Israël); Academy of Sciences, Mathematics Institute, Ztiná 25, 11567 Prague 1 (République Thèque)},
author = {Arazy, Jonathan, Engliš, Miroslav},
journal = {Annales de l’institut Fourier},
keywords = {Berezin transform; geodesic symmetry; Cartan domain; stochastic operator; -Poisson extension; disk algebra; limiting behaviour of the iterates; Bergman space; fixes holomorphic functions; convolution operators},
language = {eng},
number = {4},
pages = {1101-1133},
publisher = {Association des Annales de l'Institut Fourier},
title = {Iterates and the boundary behavior of the Berezin transform},
url = {http://eudml.org/doc/115936},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Arazy, Jonathan
AU - Engliš, Miroslav
TI - Iterates and the boundary behavior of the Berezin transform
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 4
SP - 1101
EP - 1133
AB - Let $\mu $ be a measure on a domain $\Omega $ in ${\mathbb {C}}^n$ such that the Bergman space of holomorphic functions in $L^2(\Omega ,\mu )$ possesses a reproducing kernel $K(x,y)$ and $K(x,x)&gt;0$$\forall x\in \Omega $. The Berezin transform associated to $\mu $ is the integral operator\[ Bf(y) = K(y,y)^{-1} \int _\Omega f(x)\vert K(x,y)\vert ^2 \,d\mu (x). \]The number $Bf(y)$ can be interpreted as a certain mean value of $f$ around $y$, and functions satisfying $Bf=f$ as functions having a certain mean-value property. In this paper we investigate the boundary behavior of $Bf$, the existence of functions $f$ satisfying $Bf=f$ and having prescribed boundary values, and the convergence of the iterates $B^k f$, $k\rightarrow \infty $. The best results are obtained for smoothly bounded strictly pseudoconvex domains $\Omega $ with any measure $\mu $ as above, and for bounded symmetric domains $\Omega $ and $\mu $ one of the standard rotation-invariant measures on them. We also carry out similar investigation for convolution operators $B_\mu f=f*\mu $ on a bounded symmetric domain $\Omega =G/K$ with a $K$-invariant absolutely continuous probability measure $\mu $, and study the behavior of the geodesic symmetries $\phi _a$ of $\Omega $ as $a$ tends to the boundary.
LA - eng
KW - Berezin transform; geodesic symmetry; Cartan domain; stochastic operator; -Poisson extension; disk algebra; limiting behaviour of the iterates; Bergman space; fixes holomorphic functions; convolution operators
UR - http://eudml.org/doc/115936
ER -