Thin sets and boundary behavior of solutions of the Helmholtz equation

The Martin boundary for positive solutions of the Helmholtz equation in n-dimensional Euclidean space may be identified with the unit sphere. Let v denote the solution that is represented by Lebesgue surface measure on the sphere. We define a notion of thin set at the boundary and prove that for each positive solution of the Helmholtz equation, u, there is a thin set such that ulv has a limit at Lebesgue almost every point of the sphere if boundary points are approached with respect to the Martin topology outside this thin set. We deduce a limit result for u/v in the spirit of Nagel-Stein (1984).