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

TL;DR

This paper establishes $L^2$-boundedness of gradients of single layer potentials for elliptic operators with Dini mean oscillation coefficients, linking it to geometric measure theory and rectifiability of measures.

Contribution

It generalizes deep geometric results from Riesz transforms to layer potentials associated with elliptic operators with Dini-continuous coefficients.

Findings

$L^2$-boundedness of layer potentials characterizes uniform rectifiability.

Boundedness implies rectifiability of the underlying measure or set.

Unboundedness occurs under certain measure density conditions.

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.