On the geometry of measures with density bounds in a Hölder anisotropic setting
Ignacio Tejeda
TL;DR
This work develops an anisotropic, Hölder-continuous framework for the geometry of measures with density bounds by substituting Euclidean balls with point-dependent ellipses B_Λ(X,r). It shows that if the anisotropic densities and doubling properties converge at Hölder rates and the support Σ is sufficiently flat, then Σ is a C^{1,γ} n-dimensional submanifold; without flatness, Σ decomposes into a regular C^{1,γ} part plus a singular set of Hausdorff dimension at most n−3 (for n≥3). The approach combines Λ-transforms to obtain moment and β-number decay, a robust theory of Λ-pseudo tangents yielding uniform tangents, and Reifenberg-flatness arguments to upgrade flatness into smooth parametrizations. A key outcome is the regularity of the regular set and the sharp dimension bound on the singular set, with the density Θ_Λ(μ,·) playing a central role in connecting anisotropic densities to Euclidean regularity. Overall, the paper extends DKT01-type results to an anisotropic setting, enabling Hölder rates and cone-like tangents under variable Λ-geometries.
Abstract
We study the regularity of the support of a Radon measure $μ$ 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 $μ$ is flat enough, then it is a $C^{1,γ}$ $n$-dimensional submanifold of $\mathbb R^{n+1}$, for some $γ\in (0,1)$. If the flatness assumption is dropped, then the support of $μ$ is the union of a $C^{1,γ}$ $n$-dimensional submanifold of $\mathbb R^{n+1}$ and a closed singular set that is either empty if $n\leq 2$, or has Hausdorff dimension at most $n-3$ if $n\geq 3$.