On equality of the $L^\infty$ norm of the gradient under the Hausdorff and Lebesgue measure

TL;DR

This paper proves that the supremum norm of the gradient of differentiable functions is the same whether measured with respect to Hausdorff or Lebesgue measure, with implications for convergence and function classes.

Contribution

It establishes the equality of gradient norms under different measures and shows that $W^{1, \infty}$ convergence preserves differentiability and closure of $C^1$ functions.

Findings

Gradient norms are equal under Hausdorff and Lebesgue measures.

Convergence in $W^{1, \infty}$ preserves differentiability.

$C^1$ functions form a closed subset in $W^{1, \infty}( ext{domain})$.

Abstract

Let $\Omega$ be an open subset of $\mathbb R^n$, and let $f: \Omega \to \mathbb R$ be differentiable $\mathcal H^k$-almost everywhere, for some nonnegative integer $k < n$, where $\mathcal H^k$ denotes the $k$=dimensional Hausdorff measure. We show that $\|\nabla f\|_{L^\infty (\mathcal H^k)} = \|\nabla f\|_{L^\infty(\mathcal H^n)}.$ We deduce that convergence in the Sobolev space $W^{1, \infty}$ preserves everywhere differentiability. As a further corollary, we deduce that the class $C^1 (\Omega)$ of continuously differentiable functions is closed in $W^{1, \infty}(\Omega)$.