A refined Lusin type theorem for gradients

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 $\hat{I}¼$, and we obtain that the estimate on the $L^p$ norm of $Dg$ does not depend on $\hat{I}¼(E)$, if the value of $f$ is $\hat{I}¼$-a.e. orthogonal to the decomposability bundle of $\hat{I}¼$. 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.