Journal of the
Korean Mathematical Society JKMS

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 $p \ge 1$, define the Bergman-Privalov space $(AN)^p(\nu_{\alpha})$ by \[ (AN)^p(\nu_{\alpha}) = \{ f \in H(B) : \int_{B}\{ \log(1+|f|) \}^pd\nu_{\alpha} < \infty \}. \] In this paper we prove that a function $ f \in H(B) $ is in $(AN)^p(\nu_{\alpha})$ if and only if $(1+|f|)^{-2}\{\log(1+|f|)\}^{p-2} |\tilde{\nabla}f|^2 \in L^1(\nu_{\alpha}) $ in the case $ 1 < p < \infty$, or $(1+|f|)^{-2}|f|^{-1}|\tilde{\nabla}f|^2 \in L^1(\nu_{\alpha})$ in the case $p=1$, where $\tilde{\nabla}f$ is the gradient of $f$ with respect to the Bergman metric on $B$. This is an analogous result to the characterization of the Hardy spaces by M. Stoll [18] and that of the Bergman spaces by C. Ouyang-W. Yang-R. Zhao [13].

Keywords: Bergman-Privalov spaces, Privalov spaces, Bergman spaces, Riesz measure, Hardy-Orlicz spaces

MSC numbers: Primary 32A37; Secondary 32A36