On Hardy spaces in complex ellipsoids

Hansson, Thomas

Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1477-1501.

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 $B^p=\left\{z\in {ℂ}^n:{\sum }_{j=1}^n|z_j{|}^{2p_j}\le 1\right\},p_j\in {\mathbb{Z}}^{+}$, on établit la décomposition atomique et la factorisation des fonctions de $H^1$. La dualité entre $H^1$ et $BMOA$ est également considérée.

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 $B^p=\left\{z\in {ℂ}^n:{\sum }_{j=1}^n|z_j{|}^{2p_j}\le 1\right\},p_j\in {\mathbb{Z}}^{+}$, both atomic decomposition and factorization of $H^1$-functions are established. The duality between $H^1$ and $BMOA$ is also considered.

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