Lipschitz functions with prescribed blowups at many points

Abstract

In this paper we prove generalizations of Lusin-type theorems for gradients due to Giovanni Alberti, where we replace the Lebesgue measure with any Radon measure $μ$. We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at $μ$-almost every point $x$ in any direction which is not contained in the decomposability bundle $V(μ,x)$, recently introduced by Alberti and the first named author. More precisely, we prove that it is possible to construct a Lipschitz function which attains any prescribed admissible blowup at every point except for a closed set of points of arbitrarily small measure. Here a function is an admissible blowup at a point $x$ if it is null at the origin and it is the sum of a linear function on $V(μ,x)$ and a Lipschitz function on $V(μ,x)^{\perp}$.