A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a generalized Berezin transform

Ørsted, Bent; Vargas, Jorge

Annales de l'Institut Fourier, Tome 57 (2007) no. 3, pp. 693-702.

Dans cette note, nous généralisons les résultats de notre article précédent sur l’opérateur de Casimir et sur la transformée de Berezin, en prouvant la continuité $(L^2,L^2)$ d’une transformée de Berezin généralisée associée à un problème de bifurcation pour une classe de représentations unitaires définie par des opérateurs elliptiques invariants. Nous prouvons aussi que, sous des conditions générales adéquates, cette transformée de Berezin généralisée est $(L^p,L^p)$-continue pour $1\le p\le \infty .$

In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the $(L^2,L^2)$-continuity of a generalized Berezin transform associated with a branching problem for a class of unitary representations defined by invariant elliptic operators; we also show, that under suitable general conditions, this generalized Berezin transform is $(L^p,L^p)$-continuous for $1\le p\le \infty .$

Classification : 22E46
Keywords: Discrete Series representations, branching laws, invariant elliptic operators
Mot clés : Séries discrètes, lois de bifurcation, opérateurs elliptiques invariants.

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