Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 38, 2013, 771-783

POLAR SETS AND CAPACITARY POTENTIALS IN HOMOGENEOUS SPACES

Tord Sjödin

University of Umeå, Department of Mathematics and Mathematical Statistics
S-901 87 Umeå, Sweden; tord.sjodin 'at' math.umu.se

Abstract. A set E in a space X is called a polar set in X, relative to a kernel k(x,y), if there is a nonnegative measure \sigma in X such that the potential U_k^\sigma(x) = \infty precisely when x \in E. Polar sets have been characterized in various classical cases as G_\delta-sets (countable intersections of open sets) with capacity zero. We characterize polar sets in a homogeneous space (X,d,\mu) for several classes of kernels k(x,y), among them the Riesz \alpha-kernels and logarithmic Riesz kernels. The later case seems to be new even in Rⁿ.

2010 Mathematics Subject Classification: Primary 26E25; Secondary 46B20, 49J50.

Key words: Metric space, doubling measure, Riesz kernel, definite kernel, consistent kernel, measure, potential, capacity, energy, capacitary measure, capacitary potential, polar set.

Reference to this article: T. Sjödin: Polar sets and capacitary potentials in homogeneous spaces. Ann. Acad. Sci. Fenn. Math. 38 (2013), 771-783.

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doi:10.5186/aasfm.2013.3851