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

Aline Bonami (1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

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\neq2. 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

Domaines: Analyse classique [math.CA]; Variables complexes [math.CV]

Aline Bonami. Three related problems on Bergman spaces of tube domains over symmetric cones. Harmonic Analysis on Complex Homogeneous Domains and Lie Groups, 2002, Rome, Italy. pp.183-197. ⟨hal-00087902⟩