On equality of the $L^\infty$ norm of the gradient under the Hausdorff and Lebesgue measure
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910940888301568 |
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| author | Ng, Ze-An |
| author_facet | Ng, Ze-An |
| contents | 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. As a further corollary, we deduce that the class $C^1 (Ω)$ of continuously differentiable functions is closed in $W^{1, \infty}(Ω)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08010 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On equality of the $L^\infty$ norm of the gradient under the Hausdorff and Lebesgue measure Ng, Ze-An Classical Analysis and ODEs Functional Analysis 26A16 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. As a further corollary, we deduce that the class $C^1 (Ω)$ of continuously differentiable functions is closed in $W^{1, \infty}(Ω)$. |
| title | On equality of the $L^\infty$ norm of the gradient under the Hausdorff and Lebesgue measure |
| topic | Classical Analysis and ODEs Functional Analysis 26A16 |
| url | https://arxiv.org/abs/2505.08010 |