In the recent investigations of the Dirichlet problem in an axiomatic framework we focus our interest on the following two problems: (1) the boundary behavior of the normalized PWB-solutions, originally studied by O. Frostman and generalized by Constantinescu-Cornea and Lukeš-Malý and (2) the unicity of Keldych operators formulated by J. Lukeš, given by Bliedtner-Hansen its essential contribution. The normalized PWB-solutions H f U,X on an open set U is formed by the Perron-Wiener-Brelot’s method taking the closure of U in the one-point compactified space and putting f to be 0 at the infinity, and the effect of the ideal boundary is neglected. In this article we consider above problem under the full influence of the ideal boundary and in the conclusing result we reflect that our result is a generalized version of that obtained formerly.
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- C. Constantinescu-A. Cornea: Potential theory in harmonic spaces, Grundlehr. der mat. Wiss. 158 Berlin-Heidelberg-New York 1972.Google Scholar
- O. Frostman: Les points irréguliers dans la théorie du potentiel et la citère de Wiener, Medd. Lunds Univ. Mat. Sem. 4 (1936) 1–10.Google Scholar