Boundary functions for bounded harmonic functions

Abstract

Let D be the open unit disk in the complex plane and let C be its boundary, the unit circle. If 4 E C, then by an arc at t we mean a simple arc y with one endpoint at 4 such that y { } ( D. In this paper we use the term boundazy function in the following sense. Iff is a function defined in D and b is a function defined on a set Sc: C, then we say that b is a boundary function forf if, and only if, for each 4 E S there exists an arc y at * such that f(z) approaches 0(a) as z approaches * along y. It is known that if b is a boundary function for a continuous function, then b can be made into a function of Baire class 0 or of Baire class 1 by changing its values on at most a countable set of points [4, Theorems 2, 3], [6, Theorem 6], [8, Theorem 3]. Hence b is of Baire class at most 2. Conversely, if b is a function defined on C such that b can be made into a function of Baire class 0 or 1 by changing its values on at most a countable set, then b is a boundary function for some continuous function [1, Theorem 8]. Bagemihl and Piranian gave an example [1, Theorem 6] of a harmonic function having a boundary function defined on C that is not of Baire class 0 or 1, and they asked [1, Problem 5] whether there exists a bounded harmonic function having a boundary function defined on C that is not of Baire class 0 or 1. In the present paper we answer this question by constructing the desired function. We then show that, despite this example, a boundary function for a bounded harmonic function always resembles a function of Baire class 0 or 1 in this respect: its set of discontinuity points is of the first category. We say that a functionfdefined in D has the asymptotic value a at a point ' E C if there exists an arc y at 4 such that f(z) approaches a as z approaches * along y. We say that f has general limit a at 4 if f(z) approaches a as z approaches 4 with no restrictions other than that z E D. Let Pr(6) denote the Poisson kernel; that is,