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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Polar sets on metric spaces


Authors: Juha Kinnunen and Nageswari Shanmugalingam
Journal: Trans. Amer. Math. Soc. 358 (2006), 11-37
MSC (2000): Primary 31C45, 49N60
Published electronically: August 25, 2005
MathSciNet review: 2171221
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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?


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Additional Information

Juha Kinnunen
Affiliation: Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland
Email: juha.kinnunen@oulu.fi

Nageswari Shanmugalingam
Affiliation: Department of Mathematical Sciences, P.O. Box 210025, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: nages@math.uc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-04085-7
PII: S 0002-9947(05)04085-7
Keywords: Minimizers, variational integrals, polar sets, zero capacity sets
Received by editor(s): February 27, 2003
Published electronically: August 25, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.