Advertisement

Preliminaries

  • Alexandru Aleman
  • William T. Ross
  • Nathan S. Feldman
Part of the Frontiers in Mathematics book series (FM)

Abstract

In this chapter, we set our notation and review some elementary facts about the Hardy spaces of general (simply connected) domains. Some good references for this material are [20, 22, 23, 31, 32]. For a simply connected domain Open image in new window , we say that an upper semicontinuous function u:ΩW→[−∞, ∞) is subharmonic if it satisfies the sub-mean value property. That is to say, at each point a∈Ω, there is an r>0 so that
$$ u(a) \leqslant \int_0^{2\pi } u (a + re^{i\theta } )\frac{{d\theta }} {{2\pi }}. $$
(2.1.1)
If f is analytic on Ω and p>0, then |f| is subharmonic. We say that a subharmonic function u has a harmonic majorant if there is a harmonic function U on Ω such that uU on Ω. By the Perron process for solving the classical Dirichlet problem [10, p. 200] [58, p. 118], one can show that if a subharmonic function u ≢ −∞ has a harmonic majorant, then u has a least harmonic majorant U in that uUV on Ω for all harmonic majorants V of u.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag AG 2009

Authors and Affiliations

  • Alexandru Aleman
    • 1
  • William T. Ross
    • 2
  • Nathan S. Feldman
    • 3
  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden
  2. 2.Department of Mathematics and Computer ScienceUniversity of RichmondRichmondUSA
  3. 3.Department of MathematicsWashington & Lee UniversityLexingtonUSA

Personalised recommendations