Failure of $L^2$ boundedness of gradients of single layer potentials for measures with zero low density

Abstract

Consider a totally irregular measure $\mu$ in ${\mathbb{R}}^{n+1}$, that is, the upper density ${\mathrm{limsup}}_{r\to 0}\frac{\mu (B(x,r))}{(2r{)}^n}$ is positive $\mu$-a.e.\ in ${\mathbb{R}}^{n+1}$, and the lower density ${\mathrm{liminf}}_{r\to 0}\frac{\mu (B(x,r))}{(2r{)}^n}$ vanishes $\mu$-a.e. in ${\mathbb{R}}^{n+1}$. We show that if $T_{\mu }f(x)=\int K(x,y)d\mu (y)$ is an operator whose kernel $K(\cdot ,\cdot )$ is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Hölder continuous coefficients, then $T_{\mu }$ is not bounded in $L^2(\mu )$. This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the $n$-dimensional Riesz transform.