A characterization of some classes of harmonic functions

In this paper we investigate harmonic Hardy-Orlicz; H-phi(B) and Bergman-Orlicz b(phi,alpha)(B) spaces, using an identity of Hardy-Stein type. We also extend the notion of the Lusin property by introducing (phi,alpha)-Lusin property with respect to a Stoltz domain. The main result in the paper is as follows: Let alpha is an element of[ - 1, infinity), phi be a nonnegative increasing convex function twice differentiable on (0, infinity), and u a harmonic function on the unit ball B in R-n. Then the following statements are equivalent: (a) u is an element of b(phi,alpha) (B), if alpha is an element of ( - 1, infinity). u is an element of H-phi(B) if alpha = - 1. (b) integral(B) phi ''(vertical bar u(x)vertical bar)vertical bar del u(x)vertical bar(2)(1 - vertical bar x vertical bar)(alpha+2) dV(x) < +infinity. (c) u has (phi, alpha)-Lusin property with respect to a Stoltz domain with halfangle beta, for any beta is an element of (0, pi/2). (d) u has (phi, alpha)-Lusin property with respect to a Stoltz domain with halfangle beta, for some beta is an element of (0, pi/2).