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Mediterranean Journal of Mathematics

, Volume 5, Issue 1, pp 61–76 | Cite as

A Characterization of Some Classes of Harmonic Functions

  • Stevo Stević
Article
  • 46 Downloads

Abstract.

In this paper we investigate harmonic Hardy-Orlicz \({\mathcal{H}}_\varphi (B)\) and Bergman-Orlicz b φ,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 \(\alpha \in [-1,\infty), \varphi\) be a nonnegative increasing convex function twice differentiable on (0, ∞), and u a harmonic function on the unit ball B in \({\mathbb{R}}^n\). Then the following statements are equivalent:
  1. (a)

    \(u \in b{\varphi,\alpha}(\it B), \rm{if} \alpha \in (-1, \infty).\,\, u \in {\mathcal{H}_\varphi}(\it B)\, \rm{if}\, \alpha = -1\).

     
  2. (b)

    \(\int_B \varphi^{\prime\prime}(|u(x)|)|\nabla u(x)|^2(1 - |x|)^{\alpha + 2} dV(x) < + \infty\).

     
  3. (c)

    u has (φ, α)-Lusin property with respect to a Stoltz domain with half-angle β, for any \(\beta \in (0, \pi/2 )\).

     
  4. (d)

    u has (φ, α)-Lusin property with respect to a Stoltz domain with half-angle β, for some \(\beta \in (0, \pi/2 )\).

     

Mathematics Subject Classification (2000).

Primary 31B05 

Keywords.

Harmonic functions Hardy-Orlicz space Bergman-Orlicz space Lusin property 

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Copyright information

© Birkhaueser 2008

Authors and Affiliations

  1. 1.Mathematical Institute of the Serbian Academy of ScienceBeogradSerbia

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