A characterization of some classes of harmonic functions

Abstract

In this paper we investigate harmonic Hardy-Orlicz Hφ(B) and Bergman-Orlicz bφ,α (B) spaces, using an identity of Hardy-Stein type. We also extend the notion of the Lusin property by introducing (φ, α)-Lusin property with respect to a Stoltz domain. The main result in the paper is as follows: Let α ∈ [-1,∞), φ be a nonnegative increasing convex function twice differentiable on (0, ∞), and u a harmonic function on the unit ball B in ℝn. Then the following statements are equivalent: (a) u ∈ bφ,α(B), if α ∈ (-1,∈). u ∈ Hφ(B) if α = -1. (b) ∫B φ″ (|u(x)|)|∇ u(x)|2(1 - |x|)α + 2 dV(x) < + ∞. (c) u has (φ, α)-Lusin property with respect to a Stoltz domain with half-angle β, for any β ∈ (0, π/2 ). (d) u has (φ, α)-Lusin property with respect to a Stoltz domain with half-angle β, for some β ∈ (0, π/2).