Semipolar Sets and Intrinsic Hausdorff Measure

Abstract

In a general potential-theoretic setting, results on the size of semipolar sets in terms of intrinsic measure of Hausdorff type defined by means of abstract "Green function" are established. In special situations, as for classical potential theory or Riesz potential theory, the intrinsic measure is equivalent to an ordinary Hausdorff measure of a suitable dimension.

For the space- time situations, as, for example, the heat potential theory, the intrinsic measure is equivalent to an anisotropic Hausdorff measure. For this case, our result solves an open problem from a paper by S.J.

Taylor and N.A.Watson published in 1985.

Keywords

• Heat equation
• Metric measure space
• Heat kernel
• Balayage space
• Green function
• Hausdorff measure
• Semipolar set
• Space-time process