A note on Sobolev-Lorentz Capacity and Hausdorff measure

TL;DR

This paper provides an elementary proof linking Sobolev-Lorentz capacity to Hausdorff measure, showing that zero capacity sets are null in Hausdorff measure and exist with specific dimension.

Contribution

It offers a new elementary proof connecting Sobolev-Lorentz capacity with Hausdorff measure, independent of non-linear potential theory.

Findings

Sets of zero Sobolev-Lorentz capacity are Hausdorff measure null sets.

Existence of sets with zero capacity and Hausdorff dimension exactly n-p.

Abstract

In this paper we give an elementary proof that sets of zero $p,1$-Sobolev-Lorentz capacity are $\mathcal{H}^{n-p}$-null sets independently of non-linear potential theory. We further show that there exists a set of Sobolev-Lorentz-$(p,1)$ capacity equal zero with Hausdorff dimension equal $n-p$.