A note on Dirichlet regularity on harmonic spaces (Q810201)

Let X be a locally compact Hausdorff space with a countable base. A harmonic space (X,U) is said to have an adjoint structure \(U^*\) if both (X,U) and \((X,U^*)\) are P-harmonic spaces in the sense of \textit{C. Constantinescu} and \textit{A. Cornea} [Potential theory on harmonic spaces (1972; Zbl 0248.31011)] and the associated Green function G(x,y) satisfies some auxiliary conditions. The following criterion for the regularity of the boundary point is proved. Let D be an open set in X and let \(x_ 0\in \partial D\). If \(x_ 0\) is a U-regular boundary point, then \(G(x_ 0,\cdot)\) is \(U^*\)-quasi- bounded on D. The converse assertion is valid if either \(\{x_ 0\}\) is a \(U^*\)-polar set or \(G(x_ 0,\cdot)\) is continuous on \(\bar D.\) The corresponding criterium for the parabolic periodic \(\Delta\)-\(\partial /\partial t\) is given as a corollary.