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On the base and the essential base in parabolic potential theory. (English) Zbl 0712.31001
This paper is concerned with notions of thinness for the heat equation. In section 1 several Wiener-type criteria (involving thermal outer capacity) are given for an arbitrary set in \({\mathbb{R}}^{n+1}\) to be non- thin at a point z. Section 2 deals with semi-polar sets: a set E is said to be semi-polar at z if there exists a fine neighbourhood V of z such that \(E\cap V\) is the countable union of everywhere thin sets. Several Wiener-type criteria are given for a Borel set E to be non-semi-polar at z. These now involve “continuous thermal capacity”, which, for a compact set K, is defined by \(\alpha (K)=\sup \{\mu ({\mathbb{R}}^{n+1})\}:\) here the supremum is over all non-negative Radon measures \(\mu\) with support in K such that the heat potential, \(P\mu\), of \(\mu\) is continuous on \({\mathbb{R}}^{n+1}\) and satisfies \(P\mu\leq 1\) everywhere. Finally, section 3 gives an application to the Choquet boundary.
Reviewer: S.J.Gardiner

MSC:
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
35K05 Heat equation
31B35 Connections of harmonic functions with differential equations in higher dimensions
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