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 $$\mu $$ . We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at $$\mu $$ -almost every point x in any direction which is not contained in the decomposability bundle $$V(\mu ,x)$$ , recently introduced by Alberti and the first 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 $$V(\mu ,x)$$ and a Lipschitz function on $$V(\mu ,x)^{\perp }$$ .