New characterizations for Fock spaces
[Nouvelles caractérisations pour les espaces de Fock]
Bao, Guanlong1 ; Ma, Pan2 ; Zhu, Kehe3
1Department of Mathematics, Shantou University, Shantou, Guangdong 515821 (China)
2School of Mathematics, and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083 (China)
3Department of, Mathematics and Statistics, SUNY, Albany, NY 12222 (USA)
Annales de l'Institut Fourier, Online first, 18 p.

We show that the maximal Fock space $F^\infty _\alpha $ on $\mathbb{C}^n$ is a Lipschitz space, that is, there exists a distance $d_\alpha $ on $\mathbb{C}^n$ such that an entire function $f$ on $\mathbb{C}^n$ belongs to $F^\infty _\alpha $ if and only if

\[ |f(z)-f(w)|\le Cd_\alpha (z,w) \]

for some constant $C$ and all $z,w\in \mathbb{C}^n$. This can be considered the Fock space version of the following classical result in complex analysis: a holomorphic function $f$ on the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$ belongs to the Bloch space if and only if there exists a positive constant $C$ such that $|f(z)-f(w)|\le C\beta (z,w)$ for all $z,w\in \mathbb{B}^n$, where $\beta (z,w)$ is the distance on $\mathbb{B}^n$ in the Bergman metric. We also present a new approach to Hardy–Littlewood type characterizations for $F^p_\alpha $.

Nous montrons que l’espace de Fock maximal $F^\infty _\alpha $ sur $\mathbb{C}^n$ est un espace de Lipschitz, c’est-à-dire qu’il existe une distance $d_\alpha $ sur $\mathbb{C}^n$ telle qu’une fonction entière $f$ sur $\mathbb{C}^n$ appartient à $F^\infty _\alpha $ si et seulement si

\[ |f(z)-f(w)|\le Cd_\alpha (z,w) \]

pour une constante $C$ et pour tous $z,w\in \mathbb{C}^n.$ Cela peut être considéré comme la version de l’espace de Fock du résultat classique suivant en analyse complexe : une fonction holomorphe $f$ sur la boule unité $\mathbb{B}^n$ dans $\mathbb{C}^n$ appartient à l’espace de Bloch si et seulement s’il existe une constante positive $C$ telle que $|f(z)-f(w)|\le C\beta (z,w)$ pour tous $z,w\in \mathbb{B}^n$, où $\beta (z,w)$ est la distance sur $\mathbb{B}^n$ dans la métrique de Bergman. Nous présentons également une nouvelle approche des caractérisations de type Hardy–Littlewood pour $F^p_\alpha $.

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Révisé le :
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DOI : 10.5802/aif.3773
Classification : 30H20, 46E15, 46E22
Keywords: Fock spaces, Gaussian measure, induced distance, Lipschitz space, Hardy–Littlewood type theorem
Mots-clés : Espaces de Fock, mesure gaussienne, distance induite, espace de Lipschitz, théorème de type Hardy–Littlewood

Bao, Guanlong  1   ; Ma, Pan  2   ; Zhu, Kehe  3

1 Department of Mathematics, Shantou University, Shantou, Guangdong 515821 (China)
2 School of Mathematics, and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083 (China)
3 Department of, Mathematics and Statistics, SUNY, Albany, NY 12222 (USA) 
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Bao, Guanlong; Ma, Pan; Zhu, Kehe. New characterizations for Fock spaces. Annales de l'Institut Fourier, Online first, 18 p.

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