$L^2$-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 H\"older 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.