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 $\limsup _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}$ is positive $\mu$-a.e. in ${{\mathbb {R}}}^{n+1}$, and the lower density $\liminf _{r\rightarrow 0}\frac{\mu (B(x,r))}{(2r)^n}$ vanishes $\mu$-a.e. in ${{\mathbb {R}}}^{n+1}$. We show that if $T_\mu f(x)=\int K(x,y)\,f(y)\,d\mu (y)$ 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.