Holomorphic retractions and boundary Berezin transforms

Arazy, Jonathan; Engliš, Miroslav; Kaup, Wilhelm

Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 641-657.

Dans un papier antérieur, les deux premiers co-auteurs ont démontré que la convolution d’une fonction $f$ continue sur l’adhérence d’un domaine de Cartan avec une mesure finie $\mu$ $K$-invariante dans ce domaine est aussi continue sur l’adhérence. De plus, sa restriction à chaque face $F$ de la frontière dépend uniquement de la restriction de $f$ sur $F$ et est égale à la convolution, dans $F$, de cette restriction-la, avec une certaine mesure ${\mu }_F$ sur $F$, déterminée uniquement par $\mu$. Dans cet article nous donnons une formule explicite pour ${\mu }_F$ en termes de $F$, en montrant plus particulièrement que pour des mesures $\mu$ correspondant à des transformées de Berezin, les mesures ${\mu }_F$ 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 $f$ continuous on the closure of a Cartan domain and a $K$-invariant finite measure $\mu$ on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face $F$ depends only on the restriction of $f$ to $F$ and is equal to the convolution, in $F$, of the latter restriction with some measure ${\mu }_F$ on $F$ uniquely determined by $\mu$. In this article, we give an explicit formula for ${\mu }_F$ in terms of $F$, showing in particular that for measures $\mu$ corresponding to the Berezin transforms the measures ${\mu }_F$ 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.

Classification : 32M15, 17C27, 53C35
Keywords: Berezin transform, Cartan domain, convolution operator
Mot clés : transformée de Berezin, domaine de Cartan, opérateur de convolution

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