Harmonic continuation and removable singularities in the axiomatic potential theory. (English) Zbl 0358.31007

Netuka, Ivan; Vesely, Jiri

Math. Ann. 234, 117-123 (1978).

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 J. Köhn and M. Sieveking [Revue Roumaine Math. pure. appl. 12, 1489-1502 (1967; Zbl 0158.12804)] and R. Harvey and 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 J. Bliedtner and W. Hansen [Inventiones Math. 29, 83-110 (1975; Zbl 0308.31011].

Reviewer: Ivan Netuka; Jiří Veselý

MSC:

31C05	Harmonic, subharmonic, superharmonic functions on other spaces
31C15	Potentials and capacities on other spaces

Citations:

Zbl 0158.12804; Zbl 0249.35012; Zbl 0308.31011

References:

[1]	Anger, G.: Funktionalanalytische Betrachtungen bei Differentialgleichungen unter Verwendung von Methoden der Potentialtheorie. I. Berlin: Akademie-Verlag 1967 · Zbl 0163.11901
[2]	Bauer, H.: ?ilovscher Rand und Dirichletsches Problem. Ann. Inst. Fourier11, 89-136 (1961) · Zbl 0098.06902
[3]	Bauer, H.: Harmonische R?ume und ihre Potentialtheorie. Lecture Notes in Mathematics 22. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0142.38402
[4]	Bliedtner, J., Hansen, W.: Simplicial cones in potential theory. Invent. Math.29, 83-110 (1975) · Zbl 0308.31011
[5]	Constantinescu, C., Cornea, A.: Potential theory on harmonic spaces. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0248.31011
[6]	Harvey, R., Polking, J. C.: A notion of capacity which characterizes removable singularities. Trans. Amer. Math. Soc.169, 183-195 (1972) · Zbl 0249.35012
[7]	K?hn, J., Sieveking, M.: Regul?re und extremale Randpunkte in der Potentialtheorie. Rev. Roum. Math. Pures Appl.12, 1489-1502 (1967) · Zbl 0158.12804
[8]	Kr?l, J., Luke?, J.: Indefinite harmonic continuation. ?asopis P?st. Mat.98, 87-94 (1973)
[9]	Luke?, J., Netuka, I.: The Wiener type solution of the Dirichlet problem in potential theory. Math. Ann.224, 173-178 (1976) · Zbl 0327.31009
[10]	Pradelle, A. de la: Approximation et caract?re de quasi-analyticit? dans la th?orie axiomatique des fonctions harmoniques. Ann. Inst. Fourier17, 383-399 (1967) · Zbl 0153.15501