On equality of the L^∞ norm of the gradient under the Hausdorff and Lebesgue measure

2025/05/12 by Ng, Ze-An
#26A16 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2505.08010

Abstract

Let Ω be an open subset of \mathbb Rⁿ, and let f: Ω→ \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 ‖∇ f‖_{L^∞ (\mathcal H^k)} = ‖∇ f‖_{L^∞(\mathcal Hⁿ)}. We deduce that convergence in the Sobolev space W^{1, ∞} preserves everywhere differentiability. As a further corollary, we deduce that the class C¹ (Ω) of continuously differentiable functions is closed in W^{1, ∞}(Ω).

Citations

• Notes on the Infinity-Laplace Equation

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