Potential theory of signed Riesz kernels: capacity and Hausdorff measure

Abstract

In this paper, we study the natural capacity γ α related to the Riesz kernels x/∣x∣ 1 + α in ℝ n , where 0 < α < n. For noninteger α, an unexpected behaviour arises: for 0 < α < 1, compact sets in ℝ n with finite α-Hausdorff measure have zero γ α 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 γ α capacity.