Potential theory of signed Riesz Kernels: capacity and Hausdorff measure

Abstract

In this paper, we study the natural capacity γα related to the Riesz kernels x/∣x∣1 + α in ℝn, where 0 < α < n. For noninteger α, an unexpected behaviour arises: for 0 < α < 1, compact sets in ℝn with finite α-Hausdorff measure have zero γα capacity. In the Ahlfors-David regular case, for any noninteger index α, 0 < α < n, we prove that compact sets of finite α-Hausdorff measure have zero γα capacity.