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.
From: Ze-An Ng [view email]
[v1]
Mon, 12 May 2025 19:22:37 UTC (7 KB)
[v2]
Tue, 16 Jun 2026 07:01:59 UTC (10 KB)
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