On Hardy spaces in complex ellipsoids
Annales de l'Institut Fourier, Volume 49 (1999) no. 5, pp. 1477-1501.

This paper deals with atomic decomposition and factorization of functions in the holomorphic Hardy space H 1 . Such representation theorems have been proved for strictly pseudoconvex domains. The atomic decomposition has also been proved for convex domains of finite type. Here the Hardy space was defined with respect to the ordinary Euclidean surface measure on the boundary. But for domains of finite type, it is natural to define H 1 with respect to a certain measure that degenerates near Levi-flat points and is closely related to explicit representation formulas for holomorphic functions. For the model domain B p =z n : j=1 n | z j | 2p j 1,p j + , both atomic decomposition and factorization of H 1 -functions are established. The duality between H 1 and BMOA is also considered.

Ce travail traite de la décomposition atomique et de la factorisation des fonctions de l’espace H 1 de Hardy holomorphe. Ce type de théorèmes de représentation a été démontré pour les domaines strictement pseudoconvexes. La décomposition atomique a aussi été obtenue dans les domaines convexes de type fini, lorsque l’espace de Hardy est défini à l’aide de la mesure de surface sur la frontière. Mais pour les domaines de type fini, il est naturel de définir H 1 à l’aide d’une mesure dégénérant aux points Levi-plats et étroitement liée aux formules explicites de représentation pour les fonctions holomorphes. Pour le domaine-modèle B p =z n : j=1 n | z j | 2p j 1 ,p j + , on établit la décomposition atomique et la factorisation des fonctions de H 1 . La dualité entre H 1 et BMOA est également considérée.

@article{AIF_1999__49_5_1477_0,
     author = {Hansson, Thomas},
     title = {On {Hardy} spaces in complex ellipsoids},
     journal = {Annales de l'Institut Fourier},
     pages = {1477--1501},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {5},
     year = {1999},
     doi = {10.5802/aif.1727},
     zbl = {0944.32004},
     mrnumber = {2001g:32007},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1727/}
}
TY  - JOUR
TI  - On Hardy spaces in complex ellipsoids
JO  - Annales de l'Institut Fourier
PY  - 1999
DA  - 1999///
SP  - 1477
EP  - 1501
VL  - 49
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1727/
UR  - https://zbmath.org/?q=an%3A0944.32004
UR  - https://www.ams.org/mathscinet-getitem?mr=2001g:32007
UR  - https://doi.org/10.5802/aif.1727
DO  - 10.5802/aif.1727
LA  - en
ID  - AIF_1999__49_5_1477_0
ER  - 
%0 Journal Article
%T On Hardy spaces in complex ellipsoids
%J Annales de l'Institut Fourier
%D 1999
%P 1477-1501
%V 49
%N 5
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1727
%R 10.5802/aif.1727
%G en
%F AIF_1999__49_5_1477_0
Hansson, Thomas. On Hardy spaces in complex ellipsoids. Annales de l'Institut Fourier, Volume 49 (1999) no. 5, pp. 1477-1501. doi : 10.5802/aif.1727. https://aif.centre-mersenne.org/articles/10.5802/aif.1727/

[AC] M. Andersson, H. Carlsson, Wolff type estimates and the Hp Corona problem in strictly pseudoconvex domains, Ark. Mat., 32 (1994), 255-276. | MR: 96j:32022 | Zbl: 0827.32017

[BC] A. Bonami, Ph. Charpentier, Solutions de l'équation ∂ et zéros de la classe de Nevanlinna dans certains domaines faiblement pseudoconvexes, Ann. Inst. Fourier, 32-4 (1982), 53-89. | Numdam | MR: 85f:32003 | Zbl: 0493.32005

[BL] A. Bonami, N. Lohoué, Projecteurs de Bergman et Szegő pour une classe de domaines faiblement pseudo-convexes et estimations Lp, Composito Math., 46 (1982), 159-226. | Numdam | MR: 84b:32008 | Zbl: 0538.32005

[C] M. Christ, Lectures on Singular Integral Operators, Regional conference Series in Math. N° 77 (1989). | Zbl: 0745.42008

[CRW] R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Annals of Math., 103 (1976), 611-635. | MR: 54 #843 | Zbl: 0326.32011

[CW] R.R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-643. | MR: 56 #6264 | Zbl: 0358.30023

[F] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudo-convex domains, Inventiones Math., 26 (1974), 1-66. | MR: 50 #2562 | Zbl: 0289.32012

[FS] C. Fefferman, E.M. Stein, Hp spaces of Several Variables, Acta Mathematica, 129 (1972), 137-193. | MR: 56 #6263 | Zbl: 0257.46078

[H] L. Hörmander, Lp Estimates for (Pluri-) Subharmonic Functions, Math. Scand., 20 (1967), 65-78. | Zbl: 0156.12201

[Ha] T. Hansson, On Representation Theorems for Hardy Spaces in Several Complex Variables, thesis (1997).

[KL] S.G. Krantz, S-Y. Li, Duality Theorems for Hardy and Bergman spaces on Convex Domains of Finite Type in ℂn, Ann. Inst. Fourier, 45-5 (1995), 1305-1327. | Numdam | MR: 96m:32002 | Zbl: 0835.32004

[KS] N. Kerzman, E.M. Stein, The Szegő kernel in terms of Cauchy-Fantappie kernels, Duke Math. Jour., 45, N° 2 (1978), 197-224. | MR: 58 #22676 | Zbl: 0387.32009

[McN] J. Mcneal, Estimates on the Bergman kernels of convex domains, Adv. Math., 109, N° 1 (1994), 108-139. Journal of functional analysis, 108 (1992), 361-373. | MR: 95k:32023 | Zbl: 0816.32018

[McNS] J. Mcneal, E.M. Stein, Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. Jour., 73, N° 1 (1994), 177-199. | MR: 94k:32037 | Zbl: 0801.32008

[NRSW] A. Nagel, J.P. Rosay, E.M. Stein, S. Wainger, Estimates for the Bergman and Szegő kernels in ℂ2, Ann. Math., 129 (1989), 113-149. | MR: 90g:32028 | Zbl: 0667.32016

[R1] R.M. Range, On Hölder Estimates for ∂u = f on Weakly Pseudoconvex Domains, Several Complex Variables (Cortona, 1976/1977), 247-267. | Zbl: 0421.32021

[R2] R.M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Springer-Verlag, 1986. | MR: 87i:32001 | Zbl: 0591.32002

Cited by Sources: