Three related problems of Bergman spaces of tube domains over symmetric cones. (English) Zbl 1225.32012
Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 13, No. 3-4, 183-197 (2002).
Summary: 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 [Ann. Math. (2) 94, 330–336 (1971; Zbl 0234.42009)] 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 auther and D. Békollé, G. Garrigós [IMHOTEP, J. Afr. Math. Pures Appl. 3, No. 1, 11–41 (2000; Zbl 1014.32014)]; D. Békollé, G. Garrigós and F. Ricci [Littlewood-Paley decompositions and Besov spaces related to symmetric cones. (2001; http://arxiv.org/abs/math/0305072)] and D. Békollé, M. Peloso and F. Ricci [Math. Z. 237, 31–59 (2001; Zbl 0983.32001)], 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.
In this survey, based on joint work of the auther and D. Békollé, G. Garrigós [IMHOTEP, J. Afr. Math. Pures Appl. 3, No. 1, 11–41 (2000; Zbl 1014.32014)]; D. Békollé, G. Garrigós and F. Ricci [Littlewood-Paley decompositions and Besov spaces related to symmetric cones. (2001; http://arxiv.org/abs/math/0305072)] and D. Békollé, M. Peloso and F. Ricci [Math. Z. 237, 31–59 (2001; Zbl 0983.32001)], 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.
MSC:
32A36 | Bergman spaces of functions in several complex variables |
32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |