Hausdorff measure and decay rate of Riesz capacity

TL;DR

This paper explores how the decay rate of Riesz capacity relates to Hausdorff measure, providing new insights for rectifiable sets and fractals, with implications for measure theory and geometric analysis.

Contribution

It establishes a connection between Riesz capacity decay and Hausdorff measure, including a measure-theoretic proof for subadditivity of Riesz energy.

Findings

Decay rate of Riesz capacity yields Hausdorff measure for rectifiable sets.

A one-sided decay estimate is derived for self-similar fractals.

A measure-theoretic proof of subadditivity of Riesz energy is provided.

Abstract

The decay rate of Riesz capacity as the exponent increases to the dimension of the set is shown to yield Hausdorff measure. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For strictly self-similar fractals, a one-sided decay estimate is found. Along the way, a purely measure theoretic proof is given for subadditivity of the reciprocal of Riesz energy.