On Hardy spaces in complex ellipsoids

Hansson, Thomas

Annales de l'Institut Fourier, Tome 49 (1999), p. 1477-1501 / Harvested from Numdam

Résumé

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 $B^{p}=\big \lbrace z\in {\Bbb C}^{n}: \sum _{j=1}^{n}|z_{j}|^{2p_{j}}~\le ~1 ~\big \rbrace , p_{j}\in {\Bbb Z}^{+}$, on établit la décomposition atomique et la factorisation des fonctions de $H^{1}$. La dualité entre $H^{1}$ et $BMOA$ est également considérée.

Abstract

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 $B^{p}=\big \lbrace z\in {\Bbb C}^{n}: \sum _{j=1}^{n}|z_{j}|^{2p_{j}}\le 1 \big \rbrace , p_{j}\in {\Bbb Z}^{+}$, both atomic decomposition and factorization of $H^{1}$-functions are established. The duality between $H^{1}$ and $BMOA$ is also considered.

Détail

Publié le : 1999-01-01

MR 2001g:32007 | Zbl 0944.32004

DOI : https://doi.org/10.5802/aif.1727

Bibliographie

[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