Subdifferentiable functions satisfy Lusin properties of class $C^1$ or $C^2$

Abstract

Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be a function. Assume that for a measurable set $\Omega$ and almost every $x\in \Omega$ there exists a vector ${\xi }_x\in {\mathbb{R}}^n$ such that $$\underset{h\to 0}{\mathrm{liminf}}\frac{f(x+h)-f(x)-⟨{\xi }_x,h⟩}{|h{|}^2}>-\infty .$$ Then we show that $f$ satisfies a Lusin-type property of order $2$ in $\Omega$, that is to say, for every $\epsilon >0$ there exists a function $g\in C^2({\mathbb{R}}^n)$ such that ${\mathcal{L}}^n\left(\{x\in \Omega :f(x)\ne g(x)\}\right)\le \epsilon$. In particular every function which has a nonempty proximal subdifferential almost everywhere also has the Lusin property of class $C^2$. We also obtain a similar result (replacing $C^2$ with $C^1$) for the Fréchet subdifferential. Finally we provide some examples showing that this kind of results are no longer true for "Taylor subexpansions" of higher order.

The example showing that the main results fail for $C^k$ with $k\ge 3$ has been changed. An example showing that the main results fail if we replace the Frechet subdifferential or the proximal subdifferential with the limiting subdifferential has been added