A characterization of weighted Bergman-Privalov spaces on the unit ball of $\mathbb{C}^n$

Abstract

Let B denote the unit ball in mathbb{C}^n, and nu thenormalized Lebesgue measure on B . For alpha > -1 , definednu_{alpha}(z) = c_{alpha}(1-|z|^2)^{alpha}dnu(z), z inB. Here c_{alpha} is a positive constant such thatnu_{alpha}(B) = 1. Let H(B) denote the space of allholomorphic functions in B. For p ge 1, define theBergman-Privalov space (AN)^p(nu_{alpha}) by[(AN)^p(nu_{alpha}) = { f in H(B) : int_{B}{ log(1+|f|)}^pdnu_{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 inL^1(nu_{alpha}) in the case 1 < p < infty, or(1+|f|)^{-2}|f|^{-1}|Tilde{nabla}f|^2 in L^1(nu_{alpha}) inthe case p=1, where Tilde{nabla}f is the gradient of fwith respect to the Bergman metric on B. This is an analogousresult to the characterization of the Hardy spaces by M. Stollcite{St2} and that of the Bergman spaces by C. Ouyang-W. Yang-R.Zhao cite{OYZ}.