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

DOI10.5802/JEP.211arXiv2104.04730WikidataQ122997097 ScholiaQ122997097MaRDI QIDQ6364955

Thierry De Pauw

Publication date: 10 April 2021

Abstract: Letting $Asubsetmathbb{R}^n$ be Borel measurable and $W_0:AomathbbG(n,m)$ Lipschitzian, we establish that �egin{equation*} limsup_{r o 0^+} frac{mathcal{H}^m left[ A cap B(x,r) cap (x+ W_0(x)) ight]}{alpha(m)r^m} geq frac{1}{2^n}, end{equation*} for $mathcal{L}^n$-almost every $xinA$. In particular, it follows that $A$ is $mathcal{L}^n$-negligible if and only if $mathcal{H}^m(Acap(x+W_0(x))=0$, for $mathcal{L}^n$-almost every $xinA$.

Mathematics Subject Classification ID

Length, area, volume, other geometric measure theory (28A75) Integration of real functions of several variables: length, area, volume (26B15) Abstract differentiation theory, differentiation of set functions (28A15)

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