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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
31D05 Axiomatic potential theory