Polar sets on metric spaces

Kinnunen, Juha; Shanmugalingam, Nageswari

doi:10.1090/S0002-9947-05-04085-7

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?

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