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

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. Consider a totally irregular measure μ in R n+1 , that is, the upper density lim supr→0μ(B(x,r))(2r)n is positive μ-a.e. in R n+1 , and the lower density lim infr→0μ(B(x,r))(2r)n vanishes μ-a.e. in R n+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 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.