A refined Lusin type theorem for gradients

TL;DR

This paper extends Lusin's theorem for gradients to Radon measures, showing that vector fields can be approximated by gradients of smooth functions outside small measure sets, with implications for the flat chain conjecture.

Contribution

It generalizes Lusin's theorem for gradients to arbitrary Radon measures and connects this to the flat chain conjecture in geometric measure theory.

Findings

Extension of Lusin's theorem to Radon measures.

Implication for the 1-dimensional flat chain conjecture.

Potential generalization to k-forms and higher dimensions.

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 $\mu$, and we obtain that the estimate on the $L^p$ norm of $Dg$ does not depend on $\mu(E)$, if the value of $f$ is $\mu$-a.e. orthogonal to the decomposability bundle of $\mu$. 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.