arXiv2022
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 $$ω_A(r)=\sup_{x\in \mathbb{R}^{n+1}} \frac{1}{|B(x,r)|}\int_{B(x,r)}\Big|A(z)-\frac{1}{|B(x,r)|}\int_{B(x,r)}A\Big|\,dz,$$ then, under suitable Dini-type assumptions on $ω_A$, we prove the following: if $μ$ is a compactly supported Radon measure in $\mathbb{R}^{n+1}$, $n \geq 2$, and $T_μf(x)=\int \nabla_xΓ_A (x,y)f(y)\, dμ(y)$ denotes the gradient of the single layer potential associated with $L_A$, then $$ 1+ \|T_μ\|_{L^2(μ)\to L^2(μ)}\approx 1+ \|\mathcal R_μ\|_{L^2(μ)\to L^2(μ)},$$ where $\mathcal R_μ$ indicates the $n$-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for $\mathcal R_μ$, which were recently extended to $T_μ$ associated with $L_A$ with Hölder continuous coefficients. In particular, we show the following: 1) If $μ$ is an $n$-Ahlfors-David-regular measure on $\mathbb{R}^{n+1}$ with compact support, then $T_μ$ is bounded on $L^2(μ)$ if and only if $μ$ 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 $μ\not\equiv 0$ satisfies $\limsup_{r\to 0}\tfrac{μ(B(x,r))}{(2r)^n}$ {is positive and finite} for $μ$-a.e. $x\in \mathbb{R}^{n+1}$ and $\liminf_{r\to 0}\tfrac{μ(B(x,r))}{(2r)^n}$ vanishes for $μ$-a.e. $x\in \mathbb{R}^{n+1}$, then $T_μ$ is not bounded on $L^2(μ)$. 4) If $μ$ 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 $μ|_B$ can be covered by a UR set.