## Polar sets on metric spaces

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- by Juha Kinnunen and Nageswari Shanmugalingam PDF
- Trans. Amer. Math. Soc.
**358**(2006), 11-37 Request permission

## 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?## References

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## Additional Information

**Juha Kinnunen**- Affiliation: Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland
- MR Author ID: 349676
- Email: juha.kinnunen@oulu.fi
**Nageswari Shanmugalingam**- Affiliation: Department of Mathematical Sciences, P.O. Box 210025, University of Cincinnati, Cincinnati, Ohio 45221-0025
- MR Author ID: 666716
- Email: nages@math.uc.edu
- Received by editor(s): February 27, 2003
- Published electronically: August 25, 2005
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**358**(2006), 11-37 - MSC (2000): Primary 31C45, 49N60
- DOI: https://doi.org/10.1090/S0002-9947-05-04085-7
- MathSciNet review: 2171221