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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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