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

Ze-An Ng

Published 2025-05-12Version 1

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 $W^{1, \infty}(\Omega)$.