Geodesic
Characterizations of Hardy–Orlicz and Bergman–Orlicz spaces
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 34, Tome 333 (2006), pp. 43-53 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Let $\widetilde\nabla$ и $\tau$ denote the invariant gradient and invariant measure on the unit ball $B$ of $\mathbb C^n$, respectively. Assume that $f$ is a holomorphic function on $B$ and $\varphi\in C^2 ({\mathbb R})$ is a nonnegative nondecreasing convex function. Then $f$ is in the Hardy–Orlicz space $H_\varphi(B)$ if and only if $$ \int_B\varphi''(\log|f(z)|)\frac{|\widetilde\nabla f(z)|^2}{|f(z)|^2}(1-|z|^2)^n\,d\tau(z)<\infty. $$ Analogous characterizations of Bergman–Orlicz spaces are obtained. 
@article{ZNSL_2006_333_a3,
     author = {E. Doubtsov},
     title = {Characterizations of {Hardy{\textendash}Orlicz} and {Bergman{\textendash}Orlicz} spaces},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {43--53},
     year = {2006},
     volume = {333},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a3/}
}
TY  - JOUR
AU  - E. Doubtsov
TI  - Characterizations of Hardy–Orlicz and Bergman–Orlicz spaces
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2006
SP  - 43
EP  - 53
VL  - 333
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a3/
LA  - ru
ID  - ZNSL_2006_333_a3
ER  - 
%0 Journal Article
%A E. Doubtsov
%T Characterizations of Hardy–Orlicz and Bergman–Orlicz spaces
%J Zapiski Nauchnykh Seminarov POMI
%D 2006
%P 43-53
%V 333
%U http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a3/
%G ru
%F ZNSL_2006_333_a3
E. Doubtsov. Characterizations of Hardy–Orlicz and Bergman–Orlicz spaces. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 34, Tome 333 (2006), pp. 43-53. http://geodesic.mathdoc.fr/item/ZNSL_2006_333_a3/

[1] F. L. Nazarov, N. A. Shirokov, Opisaniya prostranstva Khardi $H^2$ v kruge, Doklad na seminare, POMI, 2005

[2] Y. Matsugu, J. Miyasawa, S. Ueki, “A characterization of weighted Bergman–Privalov spaces on the unit ball of $\mathbb{C}^n$”, J. Korean Math. Soc., 39:5 (2002), 783–800 | DOI | MR | Zbl

[3] Y. Matsugu, J. Miyasawa, “A characterization of weighted Bergman–Orlicz spaces on the unit ball of $\mathbb{C}^n$”, J. Aust. Math. Soc., 74:1 (2003), 5–17 | DOI | MR | Zbl

[4] C. Ouyang, J. Riihentaus, “A characterization of Hardy–Orlicz spaces on $\mathbb{C}^n$”, Math. Scand., 80:1 (1997), 25–40 | MR | Zbl

[5] C. Ouyang, W. Yang, R. Zhao, “Characterizations of Bergman spaces and Bloch space in the unit ball of $\mathbb{C}^n$”, Trans. Amer. Math. Soc., 347:11 (1995), 4301–4313 | DOI | MR | Zbl

[6] M. Stoll, “A characterization of Hardy spaces on the unit ball of $\mathbb{C}^n$”, J. London Math. Soc. (2), 48:1 (1993), 126–136 | DOI | MR | Zbl

[7] M. Stoll, “A characterization of Hardy–Orlicz spaces on planar domains”, Proc. Amer. Math. Soc., 117:4 (1993), 1031–1038 | DOI | MR | Zbl

[8] M. Stoll, Invariant potential theory in the unit ball of $\mathbb{C}^n$, London Math. Soc. Lecture Note Series, 199, Cambridge Univ. Press, Cambridge, 1994 | MR | Zbl

[9] K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, 226, Springer-Verlag, New York, 2005 | MR