Annales Academiĉ Scientiarum Fennicĉ
Mathematica
Volumen 34, 2009, 179-194

SOBOLEV CAPACITY AND HAUSDORFF MEASURES IN METRIC MEASURE SPACES

Serban Costea

McMaster University, Department of Mathematics and Statistics
1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada; secostea 'at' math.mcmaster.ca

Abstract. This paper studies the relative Sobolev p-capacity in proper metric measure spaces when 1 < p < \infty. We prove that this relative Sobolev p-capacity is Choquet. In addition, if the space X is doubling, unbounded, admits a weak (1,p)-Poincaré inequality and has an "upper dimension" Q for some p \le Q < \infty, then we obtain lower estimates of the relative Sobolev p-capacities in terms of the Hausdorff content associated with continuous and doubling gauge functions h satisfying the decay condition

\int_0^1 (h(t) / t^Q-p)^1/p dt/t < \infty.

This condition generalizes a well-known condition in Rⁿ.

2000 Mathematics Subject Classification: Primary 31C15, 46E30.

Key words: Sobolev capacity, Hausdorff measures.

Reference to this article: S. Costea: Sobolev capacity and Hausdorff measures in metric measure spaces. Ann. Acad. Sci. Fenn. Math. 34 (2009), 179-194.

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