A characterization of weighted Bergman-Privalov spaces on the unit ball of Cn

Let B denote the unit ball in C-n, and nu the normalized Lebesgue measure on B. For alpha > -1, define dnu(alpha)(z) = c(alpha)(1 - \z\(2))(alpha)dnu(z), z is an element of B. Here c(alpha) is a positive constant such that nu(alpha)(B) = 1. Let H(B) denote the space of all holomorphic functions in B. For p greater than or equal to 1, define the Bergman-Privalov space (AN)(p)(nu(alpha)) by (AN)(p)(nu(alpha)) = {f is an element of H(B) : integral(B) {log(1 + \1\}(p) dnu(alpha) < infinity}. In this paper we prove that a function f is an element of H(B) is in (AN)(p)(nu(alpha)) if and only if (1 + \f\)(-2){log(1 + \f\)}(p-2)\delf\(2) is an element of L-1(nu(alpha)) in the case 1 < p < infinity, or (1 + \f\)(-2)\f\(-1)\delf\(2) is an element of L-1(nu(alpha)) in the case p = 1, where delf 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].