$L^2$-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

We consider a uniformly elliptic operator $L_A$ in divergence form associated with an $(n+1)\times (n+1)$-matrix $A$ with real, merely bounded, and possibly non-symmetric coefficients. If $${\omega }_A(r)=\underset{x\in {\mathbb{R}}^{n+1}}{\sup }\frac{1}{\mathrm{\mid }B(x,r)\mathrm{\mid }}{\int }_{B(x,r)}\mid A(z)-\frac{1}{\mathrm{\mid }B(x,r)\mathrm{\mid }}{\int }_{B(x,r)}A\mid dz,$$ then, under suitable Dini-type assumptions on ${\omega }_A$, we prove the following: if $\mu$ is a compactly supported Radon measure in ${\mathbb{R}}^{n+1}$, $n\ge 2$, and $T_\mu f(x)=\int \nabla_x\Gamma_A (x,y)f(y), d\mu(y)$ denotes the gradient of the single layer potential associated with $L_A$, then $$1+\mathrm{\parallel }{T}_{\mu }{\mathrm{\parallel }}_{L^2(\mu )\to L^2(\mu )}\approx 1+\mathrm{\parallel }{\mathcal{R}}_{\mu }{\mathrm{\parallel }}_{L^2(\mu )\to L^2(\mu )},$$ where ${\mathcal{R}}_{\mu }$ indicates the $n$-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for $\mathcal R_\mu$,whichwererecentlyextendedto$T_\mu$associatedwith$L_A$ with H"older continuous coefficients. In particular, we show the following: 1) If $\mu$ is an $n$-Ahlfors-David-regular measure on ${\mathbb{R}}^{n+1}$ with compact support, then $T_{\mu }$ is bounded on $L^2(\mu )$ if and only if $\mu$ is uniformly $n$-rectifiable. 2) Let $E\subset {\mathbb{R}}^{n+1}$ be compact and ${\mathcal{H}}^n(E)<\mathrm{\infty }$. If $T_{{\mathcal{H}}^n{\mathrm{\mid }}_E}$ is bounded on $L^2(\mathcal H^n|E)$,then$E$is$n$-rectifiable.3)If$\mu\not\equiv 0$satisfies$\limsup{r\to 0}\tfrac{\mu(B(x,r))}{(2r)^n}$ispositiveandfinitefor$\mu$-a\mathrm{.}e\mathrm{.}$x\in $\mathbb{R}$^{n+1}$and$\liminf_{r\to 0}\tfrac{\mu(B(x,r))}{(2r)^n}$vanishesfor$\mu$-a\mathrm{.}e\mathrm{.}$x\in $\mathbb{R}$^{n+1}$, then $T_{\mu }$ is not bounded on $L^2(\mu )$. 4) If $\mu$ is a compactly supported Radon measure satisfying a certain set of local conditions at the level of a ball $B$ with small radius, then a significant portion of $\mu {\mathrm{\mid }}_B$ can be covered by a UR set.