Factorization theorems for different classes of analytic functions in multiply connected domains

Dmitry Khavinson

Vol. 108 (1983), No. 2, 295–318

DOI: 10.2140/pjm.1983.108.295

Abstract

This paper consists of four sections. In the first section we give a survey on the reproducing kernel for harmonic functions in finitely-connected Jordan regions. We also prove a certain version of Fatou’s theorem which we will use in the next sections.

In the second part we construct the generalized Schwarz kernel for an arbitrary finitely-connected Jordan domain. This kernel reproduces any continuous single-valued analytic function inside the domain by the boundary values of its real part. Also, we give an explicit formula for the real part of this kernel in terms of the harmonic measures.

In the third section we study the Blaschke products in arbitrary Jordan domains.

The main results are contained in the fourth section. There we prove factorization theorems for the classes N, N₊, H_p and E_p.

Mathematical Subject Classification 2000

Primary: 30C40

Secondary: 30D50, 30D55

Milestones

Received: 30 May 1980

Revised: 22 November 1982

Published: 1 October 1983