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On the identity of Keldych solutions. (English) Zbl 0642.31005
This paper is motivated by the following question originally posed by J. Lukeš and recently communicated to the author by I. Netuka: Let U be a relatively compact open subset of a “nice” \({\mathcal P}\)-harmonic space \((X,^*{\mathcal H})\). Let f be a continuous real function on the boundary \(U^*\) of U such that the generalized solution \(H^ Uf\) of the Dirichlet problem and the function \(D^ Uf\) associated to the solution of the weak Dirichlet problem coincide. Does this imply that \(Af=H^ Uf\) for every Keldych operator A for U? Here a Keldych operator for U is a positive linear map A from the space \({\mathcal C}(U^*)\) of all continuous real functions on \(U^*\) into the space \({\mathcal H}(U)\) of all harmonic functions on U such that \(A(h|_{U^*})=h|_ U\) for every function \(h\in {\mathcal C}(\bar U)\) which is harmonic on U. We recall that \(H^ U\) and \(D^ U\) are Keldych operators given by \(H^ Uf(x)=\epsilon_ x^{\complement U}(f)\) and \(D^ Uf(x)=\epsilon_ x^{\beta (\complement U)}(f)\) where \(\beta\) (\(\complement U)\) is the essential base of \(\complement U.\)
As we shall see already the heat equation on \({\mathbb{R}}^ 2\) provides an example showing that the answer is negative. However, it will turn out that for every harmonic space the answer is positive if we modify the original question allowing only Keldych operators A which satisfy \(A(p|_{U^*})\leq p|_ U\) for every potential p on X.
31D05 Axiomatic potential theory
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