Rectifiability and tangents in a rough Riemannian setting

Abstract

Characterizing rectifiability of Radon measures in Euclidean space has led to fundamental contributions to geometric measure theory. Conditions involving existence of principal values of certain singular integrals \cite{mattila1995rectifiable} and the existence of densities with respect to Euclidean balls \cite{preiss1987geometry} have given rise to major breakthroughs. We study similar questions in a rough elliptic setting where Euclidean balls $B(a,r)$ are replaced by ellipses $B_{\Lambda}(a,r)$ whose eccentricity and principal axes depend on $a$. Given $\Lambda : \mathbb{R}^{n} \to GL(n,\mathbb{R})$, consider the family of ellipses $B_{\Lambda}(a,r) = a + \Lambda(a) B(0,r)$. We characterize $m$-rectifiability in terms of the almost everywhere existence of the densities $$ \theta^{m}_{\Lambda(a)}(\mu,a) = \lim_{r \downarrow 0} \frac{\mu(B_{\Lambda}(a,r))}{r^{m}} \in (0, \infty). $$ We characterize $m$-rectifiable measures in terms of the existence of the principal values-- and even under the weaker assumptions that $$ \lim_{\epsilon \downarrow 0} \int_{B_{\Lambda}(a,\epsilon R) \setminus B_{\Lambda}(a, \epsilon r)} \frac{\Lambda(a)^{-1}(y-a)}{|\Lambda(a)^{-1}(y-a)|^{m+1}} d \mu(y) = 0 \quad \forall 0 < r < R $$ when $0 < \theta^{m}_{*}(\mu,a) < \infty$ almost everywhere. We apply the second result to characterize $(n-1)$-rectifiable measures in $\mathbb{R}^{n}$ in terms of the behavior of the gradient of the single layer potential to the PDE $L_{A} u = - \textrm{div}(A \nabla u)$ under weak continuity assumptions on $A$.