On the geometry of measures with density bounds in a Hölder anisotropic setting

Abstract

We study the regularity of the support of a Radon measure $\mu$ on ${\mathbb{R}}^{n+1}$ for which anisotropic versions of its $n$-dimensional density ratio and its doubling character are assumed to converge with Hölder rate. We show that in either case, if the support of $\mu$ is flat enough, then it is a $C^{1,\gamma }$ $n$-dimensional submanifold of ${\mathbb{R}}^{n+1}$, for some $\gamma \in (0,1)$. If the flatness assumption is dropped, then the support of $\mu$ is the union of a $C^{1,\gamma }$ $n$-dimensional submanifold of ${\mathbb{R}}^{n+1}$ and a closed singular set that is either empty if $n\le 2$, or has Hausdorff dimension at most $n-3$ if $n\ge 3$.

59 pages, section 9 added, improvement to Theorem 1.3 incorporated. Feedback is welcome!