arXiv2023
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 $A$.