Failure of $$L^2$$ 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 Holder 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.