×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite