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

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 $\Vert \nabla f{\Vert }_{L^{\infty }({\mathcal{H}}^k)}=\Vert \nabla f{\Vert }_{L^{\infty }({\mathcal{H}}^n)}.$ We deduce that convergence in the Sobolev space $W^{1,\infty }$ preserves everywhere differentiability.