Three related problems of Bergman spaces of tube domains over symmetric cones

Bonami, Aline

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 13 (2002), p. 183-197 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

It has been known for a long time that the Szegö projection of tube domains over irreducible symmetric cones is unbounded in $L^{p}$ for $p \neq 2$. Indeed, this is a consequence of the fact that the characteristic function of a disc is not a Fourier multiplier, a fundamental theorem proved by C. Fefferman in the 70’s. The same problem, related to the Bergman projection, deserves a different approach. In this survey, based on joint work of the author with D. Békollé, G. Garrigós, M. Peloso and F. Ricci, we give partial results on the range of $p$ for which it is bounded. We also show that there are two equivalent problems, of independent interest. One is a generalization of Hardy inequality for holomorphic functions. The other one is the characterization of the boundary values of functions in the Bergman spaces in terms of an adapted Littlewood-Paley theory. This last point of view leads naturally to extend the study to spaces with mixed norm as well.

Détail

Publié le : 2002-12-01

MR 1984099 | Zbl 1225.32012

Bibliographie

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