Harmonic continuation and removable singularities in the axiomatic potential theory

Abstract

Let U be a relatively compact open subset of a harmonic space X and H = H(U) be the Banach space of all functions continuous on 0 and harmonic on U. This paper deals with the question of harmonic continuation of functions belonging to H across the boundary 0U of U. Simple examples show that, in general, there does not exist a function h~H having no harmonic continuation across any point of OU. It turns out, however, that there is a distinguished part ACOU such that most of functions of H (in the sense of the Baire's categories) have no continuation across any point of A while each h~H can be harmonically continued across the rest of the boundary. In some exceptional cases it may happen that A = 0, even for every connected U. Brelot spaces having this unusual property are completely described in l-8]. Under reasonable assumptions A represents a substantial part of OU and in important cases A just coincides with the ~ilov boundary of 0 with respect to H. Harmonic continuation is closely connected with the investigation of removable singularities of harmonic functions. For instance, if G C ~-." is an open set, F is a closed set and every function harmonic (in the classical sense) on G\F having a continuous extension on G admits a harmonic continuation on the whole of G, then F must be polar. It is also known that the converse holds on harmonic spaces in general. On the other hand, if X is ~2 endowed with the harmonic sheaf of solutions of the heat equation, then every continuous function which is superharmonic on the complement of a closed set F contained in the boundary of an absorbent set has a superharmonic continuation across F. This was observed in [-7]. Such an F need not be polar, hence in general polar sets do not play a role of all removable singularities of continuous (super)harmonic functions. It will be shown that in an arbitrary harmonic space, F is removable if and only if it is semipolar (see Theorem 12). Another equivalent condition is given in order to see relations to a result of [6] characterizing removable singularities of solutions of partial differential equations in terms of capacity.