Theory of Bergman Spaces in the Unit Ball of ${\mathbb{C}}^n$

Theory of Bergman Spaces in the Unit Ball of

ℂ^{n}

[Théorie des espaces de Bergman dans la boule unité de

ℂ^{n}

]

Zhao, Ruhan ; Zhu, Kehe

Mémoires de la Société Mathématique de France, no. 115 (2008) , 109 p.

Résumé
Abstract

Ces dernières années il y a eu un grand nombre de travaux sur les espaces de Bergman pondérés $A_{α}^{p}$ sur la boule unité $𝔹_{n}$ de $ℂ^{n}$ , où $0 < p < \infty$ et $α > - 1$ . Nous étendons cette étude, de manière très naturelle, au cas où $α$ est un nombre réel quelconque et $0 < p \leq \infty$ . Ce traitement unifié couvre tous les espaces de Bergman classiques, les espaces de Bésov, de Lipschitz, l’espace de Bloch, l’espace $H^{2}$ de Hardy, et celui appelé espace d’Arveson. Certains de nos résultats autour de la représentation entière, de l’interpolation complexe, des multiplicateurs de coefficients et des mesures de Carleson, sont nouveaux, y compris pour les espaces de Bergman ordinaires (non-pondérés) sur le disque unité.

There has been a great deal of work done in recent years on weighted Bergman spaces $A_{α}^{p}$ on the unit ball $𝔹_{n}$ of $ℂ^{n}$ , where $0 < p < \infty$ and $α > - 1$ . We extend this study in a very natural way to the case where $α$ is any real number and $0 < p \leq \infty$ . This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space $H^{2}$ , and the so-called Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk.

MR Zbl

DOI : 10.24033/msmf.427

Classification : 32A36, 32A18
Mots-clés : Unit ball, Bergman space, Lipschitz space, Bloch space, Arveson space, Besov space, Carleson measure, fractional derivative, integral representation, atomic decomposition, complex interpolation, coefficient multiplier

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     author = {Zhao, Ruhan and Zhu, Kehe},
     title = {Theory of {Bergman} {Spaces} in the {Unit} {Ball} of ${\mathbb{C}}^n$},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {115},
     year = {2008},
     doi = {10.24033/msmf.427},
     mrnumber = {2537698},
     zbl = {1176.32001},
     language = {en},
     url = {https://numdam.org/item/MSMF_2008_2_115__1_0/}
}

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Zhao, Ruhan; Zhu, Kehe. Theory of Bergman Spaces in the Unit Ball of ${\mathbb{C}}^n$. Mémoires de la Société Mathématique de France, Série 2, no. 115 (2008), 109 p. doi : 10.24033/msmf.427. http://numdam.org/item/MSMF_2008_2_115__1_0/

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