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 \(k\geq 1\) 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 \(\mathbf{R}^d\).