Positive harmonic functions on Lipschitz domains
Richard A. Hunt, Richard L. Wheeden · Transactions of the American Mathematical Society · 1970
The results of this paper are based on a study of certain kernel functions associated with Lipschitz domains D. These functions are related to harmonic measure in D and to the ideal boundary of D as defined by R. S. Martin (see [6]), and are analogous to the Poisson kernel.Let DczEn + i he a Lipschitz domain with a point P0 fixed.We say that u is a kernel function at Q0 e 8D if u(P) is positive and harmonic for P e D with u(P0) = 1 and u(P) vanishes continuously as P-+ Q for each Q e 3D, Q^Qo-One fundamental result of the paper is a uniform estimate for various approximations to kernel functions and another is the uniqueness of kernel functions.We use the uniform estimate to show that functions arising from several different constructions are in fact kernel functions, and the uniqueness then leads to further results.The first application is to kernels related to harmonic measure.Recall that the generalized solution of the Dirichlet problem in D for continuous boundary values f(Q) is f f(Q)K(P, Q) dcopo(Q), PeD, JdD where wp( ■ ) denotes harmonic measure in D and For a Lipschitz domain D, we will show that K(P, Q0) is the unique kernel function at Q0 and that K(P, Q) is a continuous function of Qe dD for fixed PeD.We will use these facts to discuss the general theory of R. S. Martin for Lipschitz domains.The development originated by Martin is based on kernels which are limits of quotients of Green's functions.Martin uses these kernels to define an ideal boundary A of D and a corresponding topology on D u A. For Lipschitz domains we will show that Martin's kernels are exactly the functions K(P, Q).We can then identify the ideal boundary with the Euclidean boundary and show that the Martin topology