$L^2$-boundedness of gradients of single layer potentials and uniform rectifiability

Abstract

Let $A(\cdot )$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with Hölder continuous real coefficients and let ${\mathcal{E}}_A(x,y)$ be the fundamental solution of the PDE $\mathrm{div}A(\cdot )\nabla u=0$ in ${\mathbb{R}}^{n+1}$. Let $\mu$ be a compactly supported $n$-AD-regular measure in ${\mathbb{R}}^{n+1}$ and consider the associated operator $$T_{\mu }f(x)=\int {\nabla }_x{\mathcal{E}}_A(x,y)f(y)d\mu (y).$$ We show that if $T_{\mu }$ is bounded in $L^2(\mu )$, then $\mu$ is uniformly $n$-rectifiable. This extends the solution of the codimension $1$ David-Semmes problem for the Riesz transform to the gradient of the single layer potential. Together with a previous result of Conde-Alonso, Mourgoglou and Tolsa, this shows that, given $E\subset {\mathbb{R}}^{n+1}$ with finite Hausdorff measure ${\mathcal{H}}^n$, if $T_{{\mathcal{H}}^n{|}_E}$ is bounded in $L^2({\mathcal{H}}^n{|}_E)$, then $E$ is $n$-rectifiable. Further, as an application we show that if the elliptic measure associated to the above PDE is absolute continuous with respect to surface measure, then it must be rectifiable, analogously to what happens with harmonic measure.

Minor corrections and adjustments. More detailed arguments for the applications to elliptic measure

References in corpus (5)

• The Green function estimates for strongly elliptic systems of second order
• Rectifiability of harmonic measure
• $A_{\infty }$ implies NTA for a class of variable coefficient elliptic operators
• Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries
• Tangent measures of elliptic harmonic measure and applications

Cited by in corpus (1)

• The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains