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

Abstract : 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
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Submitted on : Thursday, July 27, 2006 - 2:55:52 PM
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  • HAL Id : hal-00087902, version 1

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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⟩

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