A Lusin theorem for nonlocal gradients

Abstract

We extend the celebrated result of Alberti, stating that Borel vector fields coincide with gradients of $C^1$-functions outside of a set of arbitrary small measure. We prove that a similar statement holds true in the setting of fractional gradients and $C^{0,s}-functions. Furthermore, we establish analogous results for general nonlocal gradients with appropriate function spaces. Both statements are proven by means of the translation method, demonstrating its use for the purposes of geometric measure theory. In particular, this article illustrates how the translation method transfers rigidity phenomena from the classical gradient to a broad class of nonlocal gradients.