Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation

Abstract

Letting $A\subset {\mathbb{R}}^n$ be Borel measurable and $W_0:A\to \mathbb{G}(n,m)$ Lipschitzian, we establish that \begin{equation*} \limsup_{r \to 0^+} \frac{\mathcal{H}^m \left[ A \cap B(x,r) \cap (x+ W_0(x))\right]}{α(m)r^m} \geq \frac{1}{2^n}, \end{equation*} for ${\mathcal{L}}^n$-almost every $x\in A$. In particular, it follows that $A$ is ${\mathcal{L}}^n$-negligible if and only if ${\mathcal{H}}^m(A\cap (x+W_0(x))=0$, for ${\mathcal{L}}^n$-almost every $x\in A$.

arXiv admin note: substantial text overlap with arXiv:1904.12276