arXiv subjects
Publications and source records attributed to Emily Casey.
In this article, we prove a quantitative version of Carleson's $\varepsilon^2$ conjecture in higher dimension: we characterise those Ahlfors-David regular domains in $\mathbb{R}^{n+1}$ for which the Carleson's coefficients satisfy the so-called strong geometric lemma.
Carleson's $\varepsilon^2$-conjecture states that for Jordan domains in $\mathbb{R}^2$, points on the boundary where tangents exist can be characterized in terms of the behavior of the $\varepsilon$-function. This conjecture, which was fully resolved by Jaye, Tolsa, and Villa in 2021, established that qualitative control on the rate of decay of the Carleson $\varepsilon$-function implies the existence of tangents, up to a set of measure zero. We prove that quantitative control on the rate of decay of this function gives quantitative information on the regularity of the boundary.
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 $A$.