On the boundary behavior of the Dirichlet solutions at an irregular boundary point (Q1061899)

Let X be a \({\mathcal P}\)-harmonic space with countable base, \(X^*\) a resolutive compactification of X, \(U\subset X\) an open set and x an irregular boundary point. Let \({\mathcal N}^ U_ x\) denote the cluster set of harmonic measures at x (the precise definition is given in the paper). \textit{J. Lukeš} and \textit{J. Malý} [Math. Ann. 257, 355-366 (1981; Zbl 0461.31003)] have shown that in the case where U is relatively compact \({\mathcal N}^ U_ x\) has only four types: (1) \({\mathcal N}^ U_ x=\{\epsilon_ x\}\) (x is regular), (2) \({\mathcal N}^ U_ x=\{\epsilon_ x^{CU}\}\), (3) \({\mathcal N}^ U_ x=\{\epsilon_ x,\epsilon_ x^{CU}\}\), (4) \({\mathcal N}^ U_ x=\{t\epsilon_ x+(1- t)\epsilon_ x^{CU}\); \(0\leq t\leq 1\}\). In the present paper it is shown that considering the resolutive compactification the situation is quite similar.