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

TL;DR

This paper demonstrates that certain singular integral operators, specifically gradients of fundamental solutions for elliptic operators, fail to be bounded in L^2 for measures with zero low density, extending previous results on Riesz transforms.

Contribution

It extends the known failure of L^2 boundedness of Riesz transforms to a broader class of elliptic operator gradients for irregular measures.

Findings

Gradients of fundamental solutions are not L^2 bounded for measures with zero low density.

The result generalizes previous work on Riesz transforms to elliptic operators with H"older continuous coefficients.

The measure's irregularity is characterized by positive upper and zero lower densities almost everywhere.

Abstract

Consider a totally irregular measure $\mu$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ is positive $\mu$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density $\liminf_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ vanishes $\mu$-a.e. in $\mathbb{R}^{n+1}$. We show that if $T_\mu f(x)=\int K(x,y)\,d\mu(y)$ 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 H\"older 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.