$L^2$-bounded singular integrals on a purely unrectifiable set in ${\mathbb{R}}^d$

Abstract

We construct an example of a purely unrectifiable measure $\mu$ in ${\mathbb{R}}^d$ for which the singular integrals associated to the kernels $K(x)=\frac{P_{2k+1}(x)}{|x{|}^{2k+d}}$, 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 $\frac{x}{|x{|}^d}$ in ${\mathbb{R}}^d$.