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

Abstract

Let $Ω$ be an open subset of $\mathbb R^n$, and let $f: Ω\to \mathbb R$ 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.