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The Ninomiya operators and the generalized Dirichlet problem in potential theory. (English) Zbl 0622.31006
This article is devoted to the uniqueness of the Ninomiya operators. Let X be a $${\mathfrak P}$$-harmonic space with countable base. The corresponding harmonic sheaf is denoted by $${\mathfrak H}$$ and the cone of continuous potentials on X by $${\mathfrak P}$$. Let U’$$\subset X$$ be a nonempty, relatively compact open set and let $$U_ i$$ denote the set of irregular points of U. Denote further by $${\mathcal F}(U)$$ the space of real-valued functions on U, by $${\mathfrak S}(U)$$ the cone of superharmonic functions on U, $${\mathfrak H}(U):={\mathfrak S}(U)\cap (-{\mathfrak S}(U))$$, $$H(U):=\{h\in {\mathcal C}(\bar U)|$$ $$h_{| U}\in {\mathfrak H}(U)\}$$, $$P(U):=\{p_{| \bar U}|$$ $$p\in {\mathfrak P}$$ with its superharmonic carrier $$\subset X\setminus U\}$$ and $$Q(U):=P(U)-P(U)$$. An operator $$A: {\mathcal C}(\partial U)\to {\mathcal F}(U)$$ is called a Ninomiya operator (resp. a weak Ninomiya operator) on U, if (1) A is linear and positive, (2) $$A(p_{| \partial U})=p_{| U}$$ whenever $$p\in P(U)$$ and (3) there is a strict potential $$q\in {\mathfrak P}$$ such that $$A(q_{| \partial U})\in {\mathfrak H}(U)$$, resp. $$\in -{\mathfrak S}(U).$$
The main result now reads as Theorem: Suppose that Q(U) linearly separates the points of $$\bar U$$ and contains a strictly positive function. Then the following are equivalent: a) there exists a unique weak Ninomiya operator on U, b) there exists a unique Ninomiya operator on U, c) $$U_ i$$ is negligible. Finally, a $${\mathfrak P}$$-harmonic Bauer space Y with a countable base and a nonempty, relatively compact open set $$U\subset Y$$ is constructed such that $$U_ i$$ is negligible and there exist two distinct Ninomiya (in fact Keldyš) operators on U.
Reviewer: I.Laine
MSC:
 31D05 Axiomatic potential theory