A refined Lusin type theorem for gradients

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 C1 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 Lp 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 Rnand we state a suitable generalization for k-forms, which would imply the validity of the conjecture in full generality. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.