Geodesic
Three related problems of Bergman spaces of tube domains over symmetric cones
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 13 (2002) no. 3-4, pp. 183-197.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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. 
@article{RLIN_2002_9_13_3-4_a1,
     author = {Bonami, Aline},
     title = {Three related problems of {Bergman} spaces of tube domains over symmetric cones},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {183--197},
     publisher = {mathdoc},
     volume = {Ser. 9, 13},
     number = {3-4},
     year = {2002},
     zbl = {1225.32012},
     mrnumber = {1311766},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2002_9_13_3-4_a1/}
}
TY  - JOUR
AU  - Bonami, Aline
TI  - Three related problems of Bergman spaces of tube domains over symmetric cones
JO  - Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
PY  - 2002
SP  - 183
EP  - 197
VL  - 13
IS  - 3-4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RLIN_2002_9_13_3-4_a1/
LA  - en
ID  - RLIN_2002_9_13_3-4_a1
ER  - 
%0 Journal Article
%A Bonami, Aline
%T Three related problems of Bergman spaces of tube domains over symmetric cones
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 2002
%P 183-197
%V 13
%N 3-4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RLIN_2002_9_13_3-4_a1/
%G en
%F RLIN_2002_9_13_3-4_a1
Bonami, Aline. Three related problems of Bergman spaces of tube domains over symmetric cones. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 13 (2002) no. 3-4, pp. 183-197. http://geodesic.mathdoc.fr/item/RLIN_2002_9_13_3-4_a1/

[1] D. Békollé - A. Bonami, Estimates for the Bergman and Szegö projections in two symmetric domains of $\mathbb{C}^{n}$. Colloq. Math., 68, 1995, 81-100. | fulltext EuDML | MR | Zbl

[2] D. Békollé - A. Bonami, Analysis on tube domains over light cones : some extensions of recent results. Actes des Rencontres d’Analyse Complexe: Mars 1999, Univ. Poitiers. Ed. Atlantique et ESA CNRS 6086, 2000. | Zbl

[3] D. Békollé - A. Bonami - G. Garrigós, Littlewood-Paley decompositions related to symmetric cones. IMHOTEP, to appear; available at http://www.harmonic-analysis.org | MR | Zbl

[4] D. Békollé - A. Bonami - G. Garrigós - F. Ricci, Littlewood-Paley decompositions and Besov spaces related to symmetric cones. Univ. Orléans, preprint 2001; available at http://www.harmonic-analysis.org | fulltext mini-dml

[5] D. Békollé - A. Bonami - M. Peloso - F. Ricci, Boundedness of weighted Bergman projections on tube domains over light cones. Math. Z., 237, 2001, 31-59. | DOI | MR | Zbl

[6] D. Békollé - A. Temgoua Kagou, Reproducing properties and $L^{p}$-estimates for Bergman projections in Siegel domains of type II. Studia Math., 115 (3), 1995, 219-239. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[7] R. Coifman - R. Rochberg, Representation theorems for holomorphic functions and harmonic functions in $L^{p}$. Asterisque, 77, 1980, 11-66. | MR | Zbl

[8] J. Faraut - A. Korányi, Analysis on symmetric cones. Clarendon Press, Oxford 1994. | MR | Zbl

[9] C. Fefferman, The multiplier problem for the ball. Ann. of Math., 94, 1971, 330-336. | MR | Zbl

[10] G. Garrigós, Generalized Hardy spaces on tube domains over cones. Colloq. Math., 90, 2001, 213-251. | DOI | MR | Zbl

[11] E. Stein, Some problems in harmonic analysis suggested by symmetric spaces and semi-simple Lie groups. Actes, Congrès intern. math., 1, 1970, 173-189. | MR | Zbl