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A note on Sobolev-Lorentz Capacity and Hausdorff measure

Daniel Campbell

TL;DR

This work gives an elementary, potential-theory-free proof that $\gamma_{p,1}(E)=0$ implies $\mathcal{H}^{n-p}(E)=0$ for $p\in(1,n]$, by leveraging Orlicz-Sobolev methods via MSZ results. It constructs a decomposed energy scheme leading to an Orlicz control that forces the $n-p$ Hausdorff measure to vanish, and it provides Cantor-type constructions showing that the bound is sharp, with $\dim_{\mathcal{H}}(E)=n-p$ yet $\mathcal{H}^{n-p}(E)=0$. The paper also demonstrates sharpness in the opposite direction: the result does not extend to $q>1$, through explicit counterexamples. Together, these results clarify the precise size properties of $p,1$-Sobolev-Lorentz null-sets and their relation to Hausdorff dimension, offering a more accessible route to key capacity–measure connections.

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 $n-p$.

A note on Sobolev-Lorentz Capacity and Hausdorff measure

TL;DR

This work gives an elementary, potential-theory-free proof that implies for , by leveraging Orlicz-Sobolev methods via MSZ results. It constructs a decomposed energy scheme leading to an Orlicz control that forces the Hausdorff measure to vanish, and it provides Cantor-type constructions showing that the bound is sharp, with yet . The paper also demonstrates sharpness in the opposite direction: the result does not extend to , through explicit counterexamples. Together, these results clarify the precise size properties of -Sobolev-Lorentz null-sets and their relation to Hausdorff dimension, offering a more accessible route to key capacity–measure connections.

Abstract

In this paper we give an elementary proof that sets of zero -Sobolev-Lorentz capacity are -null sets independently of non-linear potential theory. We further show that there exists a set of Sobolev-Lorentz- capacity equal zero with Hausdorff dimension equal .

Paper Structure

This paper contains 4 sections, 6 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main']}
  4. Examples showing the sharpness of the result

Key Result

Theorem 1.2

Let $p\in(1,n]$ and let $E\subset \mathbb R^n$ be such that $\gamma_{p,1}(E) = 0$. Then $\mathcal{H}^{n-p}(E) = 0$.

Figures (1)

  • Figure 1: The first three generations of the construction of a Cantor set in dimension 1 as the limit of finite sums $c_w$ and an illustration of the sets $A_w$ in generation one and two.

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • proof
  • ...and 5 more