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arXiv · 1612.05280

Lipschitz functions with prescribed blowups at many points

Andrea Marchese·Andrea Schioppa
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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 $V(\mu,x)^{\perp}$.

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Andrea Marchese, Andrea Schioppa. 2016-12-15. Lipschitz functions with prescribed blowups at many points. https://arxiv.org/abs/1612.05280

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