On the boundary values of harmonic functions

Abstract

0. A classical result of Fatou states that if u(z) is harmonic and bounded in lzl < 1 then u has a nontangential limit at almost every point ei?. The same conclusion holds if u is only bounded from below. These results have a local analogue -namely, if u(z) is harmonic in lzl < 1 and at each point ei0 of a measurable set E there is some cone in which u(z) is bounded from either above or below then u(z) has a nontangential limit at almost every eiO & E. With the aid of conformal mapping one can show the above results hold for regions of the plane more general than lzl < 1. Methods similar to those used for lzl < 1 may be applied to functions which are harmonic in the unit ball of Euclidean (n + 1)-space. However, since we lack a conformal mapping theorem for more general domains DcE +1 the situation there is more technically complicated. Known results for certain kinds of domains D c E, + 1 are due to Brelot and Doob [2], Calderon [3], Carleson [4] and Widman [7], and the purpose of this paper is to obtain nontangential boundary values for functions which are harmonic in still more general types of domains D' En + 1 and bounded from above or below in cones. The domains which we consider are all regular domains for the solution of the Dirichlet problem. For a regular domain D, our result may be stated as follows. Let EC aD and suppose for each Q E E there is a cone with vertex Q which is exterior to D. Then if u is harmonic in D and is bounded from above or below in a cone with vertex Q for each Q E E, u has a finite nontangential limit at each Q E E except for a set of harmonic measure zero. It is natural that the exceptional set be one of harmonic measure zero since if Ec AD is any set of harmonic measure zero it is easy to construct a positive harmonic function with boundary value +oo at each point of E. Our main result is a consequence of the basic result that if u is harmonic and bounded in a starlike Lipschitz domain D, then u has a finite nontangential limit at each Q E AD except for a set of harmonic measure zero. In ?1 of this paper we define the terms starlike Lipschitz domain, nontangential limit, and harmonic measure. We also include some elementary consequences of these definitions and some essential theorems on the differentiation of integrals.