The Bergman number of a plane domain

Source abstract

Let D be a domain in the complex plane C. The Hardy number of D, which was first introduced by Hansen, is the maximal number h(D) in [0,+∞] such that f belongs to the classical Hardy space Hp(D) whenever 0<p<h(D) and f is holomorphic on the unit disk D with values in D. As an analogue notion to the Hardy number of a domain D in C, we introduce the Bergman number of D, and we denote it by b(D). Our main result is that if D is regular, then h(D)=b(D). This generalizes earlier work by the author and Karamanlis for simply connected domains. The Bergman number b(D) is the maximal number in [0,+∞] such that f belongs to the weighted Bergman space Aαp(D) whenever p>0 and α>−1 satisfy 0<p α+2<b(D) and f is holomorphic on D with values in D. We also establish several results about Hardy spaces and weighted Bergman spaces, and we give a new characterization of the Hardy number and thus of the Bergman number of a regular domain with respect to the harmonic measure.