SOBOLEV CAPACITY AND HAUSDORFF MEASURES IN METRIC MEASURE SPACES

This paper studies the relative Sobolev p-capacity in proper metric measure spaces when 1 < p < infinity. We prove that, this relative Sobolev 7,,capacity is Choquet. In addition, if the space X is doubling, unbounded, admits a weak (1,p)-Poincare inequality and has an "upper dimension" Q for some p <= Q < infinity, 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 (1) integral(1)(0)(h(t)/t(Q)-p)(1/p)dt/t < infinity. This condition generalizes a well-known condition in R(n).