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

##### 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
Full Text:
##### References:
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