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Separation of points by classes of harmonic functions. (English) Zbl 0788.31006
Two subsets of \(\mathbb{R}^ n\) are called equivalent if one is the direct image of the other under (i) a linear fractional transformation if \(n=2\), (ii) an isometry if \(n \geq 3\). A domain \(D\) is called a Denjoy domain if \(\mathbb{R}^ n \backslash D \subseteq \mathbb{R}^{n-1} \times \{0\}\). The main result states that the class of all positive harmonic functions on a Green domain \(\Omega\) separates the points of \(\Omega\), if and only if \(\Omega\) is not equivalent to a Denjoy domain whose Martin compactification is homeomorphic to the Alexandroff compactification of \(\mathbb{R}^ n\). Those domains for which certain other classes of harmonic functions separate the points, are also characterized.
MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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