Harmonic continuation and removable singularities in the axiomatic potential theory
Let \(U\) be a relatively compact open subset of a harmonic space \(X\) and \(S\) be the set of all functions continuous on \(\bar{U}\) and superharmonic on \(U\). The Šilov boundary of \(U\) with respect to \(S\) will be denoted by \(\partial_S\bar{U}\) and \(H = S \cap (-S)\) will be considered as a Banach space (\(H\) is equipped with the supremum norm). Suppose that \(x \in \partial U\) and \(h \in H\). Then \(x\) is termed a point of continuability of \(h\), if there is a function \(h_1\) harmonic on a neighborhood \(V\) of \(x\) such that \(h=h_1\) on \(V \cap U\). Assuming that the points of \(\partial U\) are polar, the typical result concerning the possibility of harmonic continuation reads as follows: Each point of \(\partial U\setminus \partial_S \bar{U}\) is a point of continuability of any \(h\in H\) and the set of all functions of \(H\), for which no point of \(\partial_s \bar{U}\) is a point of continuability, is a dense \(G_{\delta}\) in \(H\). Suppose that \(F\subset X\) is a closed set. We say that \(F\) has \(c\)-capacity zero provided there is no non-trivial potential with support in \(F\). Further \(\beta(F)\) is the essential base of \(F\). The following theorem on removable singularities for continuous (super) harmonic functions is proved. Conditions (i) - (v) are equivalent: (i) F is semi-polar. (ii) (respectively (iii)) If \(G\) is an open set and \(f\) is a continuous function on \(G\) and superharmonic (respectively harmonic) on \(G\setminus F\), then \(f\) is superharmonic (respectively harmonic) on \(G\). (iv) \(F\) has \(c\)-capacity zero. (v) \(\beta(F)=\emptyset\). This theorem represents a generalization of some results of \textit{J. Köhn} and \textit{M. Sieveking} [Revue Roumaine Math. pure. appl. 12, 1489-1502 (1967; Zbl 0158.12804)] and \textit{R. Harvey} and \textit{J. C. Polking} [Trans. Amer. math. Soc. 169, 183-195 (1972; Zbl 0249.35012)]. Proofs of the theorems mentioned above use the theory of simplicial cones developed by \textit{J. Bliedtner} and \textit{W. Hansen} [Inventiones Math. 29, 83-110 (1975; Zbl 0308.31011].
scientific article; zbMATH DE number 3558182
scientific article