Polar sets on metric spaces

Juha Kinnunen*, Nageswari Shanmugalingam

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

13 Citations (Scopus)

Abstract

We show that if X is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of X with zero p-capacity are precisely the p-polar sets; that is, a relatively compact subset of a domain in X is of zero p-capacity if and only if there exists a p-superharmonic function whose set of singularities contains the given set. In addition, we prove that if X is a p-hyperbolic metric space, then the p-superharmonic function can be required to be p-superharmonic on the entire space X. We also study the the following question: If a set is of zero p-capacity, does there exist a p-superharmonic function whose set of singularities is precisely the given set?

Original languageEnglish
Pages (from-to)11-37
Number of pages27
JournalTransactions of the American Mathematical Society
Volume358
Issue number1
DOIs
Publication statusPublished - 2006
MoE publication typeA1 Journal article-refereed

Keywords

  • Minimizers
  • Polar sets
  • Variational integrals
  • Zero capacity sets

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