Lipschitz functions with prescribed blowups at many points

TL;DR

This paper generalizes Lusin-type theorems for gradients with respect to Radon measures, constructing Lipschitz functions with prescribed blowups at many points, extending previous non-differentiability results.

Contribution

It introduces a method to construct Lipschitz functions with arbitrary prescribed blowups at almost every point, beyond classical differentiability constraints, for general Radon measures.

Findings

Constructed Lipschitz functions with prescribed blowups at almost all points.

Extended non-differentiability results to general Radon measures.

Demonstrated the flexibility of Lipschitz functions in prescribed local behavior.

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 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 $V(\mu,x)$ and a Lipschitz function on $V(\mu,x)^{\perp}$.