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arXiv · 2411.15012

A refined Lusin type theorem for gradients

Luigi De Masi·Andrea Marchese
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Abstract

We prove a refined version of the celebrated Lusin type theorem for gradients by Alberti, stating that any Borel vector field $f$ coincides with the gradient of a $C^1$ function $g$, outside a set $E$ of arbitrarily small Lebesgue measure. We replace the Lebesgue measure with any Radon measure $μ$, and we obtain that the estimate on the $L^p$ norm of $Dg$ does not depend on $μ(E)$, if the value of $f$ is $μ$-a.e. orthogonal to the decomposability bundle of $μ$. We observe that our result implies the 1-dimensional version of the flat chain conjecture by Ambrosio and Kirchheim on the equivalence between metric currents and flat chains with finite mass in $\mathbb{R}^n$ and we state a suitable generalization for $k$-forms, which would imply the validity of the conjecture in full generality.

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Luigi De Masi, Andrea Marchese. 2024-11-22. A refined Lusin type theorem for gradients. https://arxiv.org/abs/2411.15012

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