[Rétractions holomorphiques et transformées de Berezin sur les frontières]
Dans un papier antérieur, les deux premiers co-auteurs ont démontré que la convolution d’une fonction continue sur l’adhérence d’un domaine de Cartan avec une mesure finie -invariante dans ce domaine est aussi continue sur l’adhérence. De plus, sa restriction à chaque face de la frontière dépend uniquement de la restriction de sur et est égale à la convolution, dans , de cette restriction-la, avec une certaine mesure sur , déterminée uniquement par . Dans cet article nous donnons une formule explicite pour en termes de , en montrant plus particulièrement que pour des mesures correspondant à des transformées de Berezin, les mesures correspondent à nouveau à des transformées de Berezin mais avec un décalage dans la valeur du paramètre de Wallach. Enfin, nous obtenons aussi une description simple et jolie d’une rétraction holomorphique sur ces domaines qui découle de la limite à la frontière de symétries géodésiques.
In an earlier paper, the first two authors have shown that the convolution of a function continuous on the closure of a Cartan domain and a -invariant finite measure on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face depends only on the restriction of to and is equal to the convolution, in , of the latter restriction with some measure on uniquely determined by . In this article, we give an explicit formula for in terms of , showing in particular that for measures corresponding to the Berezin transforms the measures again correspond to Berezin transforms, but with a shift in the value of the Wallach parameter. Finally, we also obtain a nice and simple description of the holomorphic retraction on these domains which arises as the boundary limit of geodesic symmetries.
Keywords: Berezin transform, Cartan domain, convolution operator
Mot clés : transformée de Berezin, domaine de Cartan, opérateur de convolution
Arazy, Jonathan 1 ; Engliš, Miroslav 2 ; Kaup, Wilhelm 3
@article{AIF_2009__59_2_641_0,
author = {Arazy, Jonathan and Engli\v{s}, Miroslav and Kaup, Wilhelm},
title = {Holomorphic retractions and boundary {Berezin} transforms},
journal = {Annales de l'Institut Fourier},
pages = {641--657},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {59},
number = {2},
year = {2009},
doi = {10.5802/aif.2444},
mrnumber = {2521432},
zbl = {1176.47026},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2444/}
}
TY - JOUR AU - Arazy, Jonathan AU - Engliš, Miroslav AU - Kaup, Wilhelm TI - Holomorphic retractions and boundary Berezin transforms JO - Annales de l'Institut Fourier PY - 2009 SP - 641 EP - 657 VL - 59 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2444/ DO - 10.5802/aif.2444 LA - en ID - AIF_2009__59_2_641_0 ER -
%0 Journal Article %A Arazy, Jonathan %A Engliš, Miroslav %A Kaup, Wilhelm %T Holomorphic retractions and boundary Berezin transforms %J Annales de l'Institut Fourier %D 2009 %P 641-657 %V 59 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2444/ %R 10.5802/aif.2444 %G en %F AIF_2009__59_2_641_0
Arazy, Jonathan; Engliš, Miroslav; Kaup, Wilhelm. Holomorphic retractions and boundary Berezin transforms. Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 641-657. doi : 10.5802/aif.2444. https://aif.centre-mersenne.org/articles/10.5802/aif.2444/
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