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

Consider a totally irregular measure μ in Rn+1, that is, the upper density lim supr→0μ(B(x,r))(2r)n is positive μ-a.e. in Rn+1, and the lower density lim infr→0μ(B(x,r))(2r)n vanishes μ-a.e. in Rn+1. We show that if Tμf(x)=∫K(x,y)f(y)dμ(y) is an operator whose kernel K(·,·) is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with Hölder continuous coefficients, then Tμ is not bounded in L2(μ). This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the n-dimensional Riesz transform.