L^2-bounded singular integrals on a purely unrectifiable set in R^d

Abstract

We construct an example of a purely unrectifiable measure $\mu$ in ${\mathbf{R}}^d$ for which the singular integrals associated to the kernels $K(x)=P_{2k+1}(x)/|x{|}^{2k+d}$, with $k\ge 1$ and $P_{2k+1}$ a homogeneous harmonic polynomial of degree $2k+1$, are bounded in $L^2(\mu )$. This contrasts starkly with the results concerning the Riesz kernel $x/|x{|}^d$ in ${\mathbf{R}}^d$