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Potential Analysis

, Volume 51, Issue 1, pp 49–69 | Cite as

Semipolar Sets and Intrinsic Hausdorff Measure

  • Wolfhard HansenEmail author
  • Ivan Netuka
Article
  • 52 Downloads

Abstract

Given a “Green function” G on a locally compact space X with countable base, a Borel set A in X is called G-semipolar, if there is no measure ν ≠ 0 supported by A such that \(G\nu :=\int G(\cdot ,y)\,d\nu (y)\) is a continuous real function on X. Introducing an intrinsic Hausdorff measuremG using G-balls B(x, ρ) := {yX : G(x, y) > 1/ρ}, it is shown that every set A in X with \(m_{G}(A)<\infty \) is contained in a G-semipolar Borel set. This is of interest, since G-semipolar sets are semipolar in the potential-theoretic sense (countable unions of totally thin sets, hit by a corresponding process at most countably many times), if G is a genuine Green function. The result has immediate consequences for classical potential theory, Riesz potentials and the heat equation (where it solves an open problem). More generally, it is applied to metric measure spaces (X, d, μ), where a continuous heat kernel with upper and lower bounds of the form tα/βΦj(d(x,y)t− 1/β), j = 1, 2, is given. Then the intrinsic Hausdorff measure on X is equivalent to an ordinary Hausdorff measure mαβ. For the corresponding space-time structure on X × ℝ, the intrinsic Hausdorff measure turns out to be equivalent to an anisotropic Hausdorff measure mα,β.

Keywords

Heat equation Metric measure space Heat kernel Balayage space Green function Hausdorff measure Semipolar set Space-time process 

Mathematics Subject Classification (2010)

28A78 31C15 31E05 31C12 31D05 35J08 35K08 60J45 60J60 60J75 

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Faculty of Mathematics and Physics, Mathematical InstituteCharles UniversityPraha 8Czech Republic

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