Preliminaries

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 , 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 u≤U 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 u≤U≤V on Ω for all harmonic majorants V of u.