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.