A CHARACTERIZATION OF WEIGHTED BERGMAN-PRIVALOV SPACES ON THE UNIT BALL OF Cn

Abstract

Let B denote the unit ball in , and ν the normalized Lebesgue measure on B. For > -1, define (z) ＝ dν(z), z B. Here is a positive constant such that (B) ＝ 1. Let H(B) denote the space of all holomorphic functions in B. For , define the Bergman-Privalov space by = : $\int_B{log(1+\midf\mid)}^pdv_\alpha\; is in if and only if in the case 1＜p＜, or in the case p ＝ 1, where 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].