Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces

Abstract

Let \(\tilde \nabla \) and τ denote the invariant gradient and invariant measure on the unit ball B of ℂn, respectively. Assume that f is a holomorphic function on B and ϕ ∈ C2(ℝ) is a nonnegative, nondecreasing, convex function. Then f belongs to the Hardy-Orlicz space H ϕ(B>) if and only if $$\int\limits_B {\varphi ''(\log \left| {f(z)} \right|)\frac{{\left| {\tilde \nabla f(z)} \right|^2 }}{{\left| {f(z)} \right|^2 }}} (1 - \left| z \right|^2 )^n d\tau (z) < \infty .$$ Analogous characterizations of Bergman-Orlicz spaces are obtained. Bibliography: 9 titles.