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

Abstract

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 $$\omega_A(r)=\sup_{x\in \mathbb{R}^{n+1}} \frac{1}{|B(x,r)|}\int_{B(x,r)}\Big|A(z)-\frac{1}{|B(x,r)|}\int_{B(x,r)}A\Big|\,dz,$$ then, under suitable Dini-type assumptions on $\omega_A$, we prove the following: if $\mu$ is a compactly supported Radon measure in $\mathbb{R}^{n+1}$, $n \geq 2$, 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 $$ 1+ \|T_\mu\|_{L^2(\mu)\to L^2(\mu)}\approx 1+ \|\mathcal R_\mu\|_{L^2(\mu)\to L^2(\mu)},$$ 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 $\mathcal R_\mu$, which were recently extended to $T_\mu$ associated with $L_A$ 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 $\mathcal H^n(E)<\infty$. If $T_{\mathcal H^n|_E}$ is bounded on $L^2(\mathcal H^n|_E)$, then $E$ is $n$-rectifiable. 3) If $\mu\not\equiv 0$ satisfies $\limsup_{r\to 0}\tfrac{\mu(B(x,r))}{(2r)^n}$ {is positive and finite} for $\mu$-a.e. $x\in \mathbb{R}^{n+1}$ and $\liminf_{r\to 0}\tfrac{\mu(B(x,r))}{(2r)^n}$ vanishes for $\mu$-a.e. $x\in \mathbb{R}^{n+1}$, 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 $\mu|_B$ can be covered by a UR set.