J. Aust. Math. Soc.  74 (2003), 5-17
A characterization of weighted Bergman-Orlicz spaces on the unit ball in $\mathbb{C}^n$

Yasuo Matsugu
  Department of Mathematical Sciences
  Faculty of Science
  Shinshu University
  390-8621 Matsumoto
  Japan
  matsugu@math.shinshu-u.ac.jp
and
Jun Miyazawa
  Department of Mathematical Sciences
  Faculty of Science
  Shinshu University
  390-8621 Matsumoto
  Japan
 


Abstract
Let $B$ denote the unit ball in $\mathbb{C}^n$, and $\nu$ the normalized Lebesgue measure on $B$. For $\alpha>-1$, define $d\nu_{\alpha}(z)  =c_{\alpha}(1-|z|^2)^{\alpha}d\nu(z)$, $z\in B$. Here $c_{\alpha}$ is a positive constant such that $\nu_{\alpha}(B)=1$. Let $H(B)$ denote the space of all holomorphic functions in $B$. For a twice differentiable, nondecreasing, nonnegative strongly convex function $\varphi$ on the real line $\mathbb{R}$, define the Bergman-Orlicz space $A_{\varphi}(\nu_{\alpha})$ by
\[ A_{\varphi}(\nu_{\alpha})=\left\{f\in H(B):
\int_B \varphi(\log|f|)\, d\nu_{\alpha}<\infty\right\}.\]
In this paper we prove that a function $f\in H(B)$ is in $A_{\varphi}(\nu_{\alpha})$ if and only if
\[\int_B \varphi''\big(\log|f(z)|\big)
\frac{|{\mathcal R}  f(z)|^2}{|z|^2|f(z)|^2}
\big(1-|z|^2\big)^2\, d\nu_{\alpha}(z)<\infty,\]
where ${\mathcal R} f(z)=\sum^n_{j=1}z_j{\partial f(z)}/{\partial  z_j}$ is the radial derivative of $f$.
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