Failure of $L^2$ boundedness of gradients of single layer potentials for measures with zero low density
José M. Conde‐Alonso, Mihalis Mourgoglou, Xavier Tolsa · arXiv (Cornell University) · 2018
Consider a totally irregular measure $μ$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{μ(B(x,r))}{(2r)^n}$ is positive $μ$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density $\liminf_{r\to0}\frac{μ(B(x,r))}{(2r)^n}$ vanishes $μ$-a.e. in $\mathbb{R}^{n+1}$. We show that if $T_μf(x)=\int K(x,y)\,dμ(y)$ 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 $T_μ$ is not bounded in $L^2(μ)$. This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the $n$-dimensional Riesz transform.