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

TL;DR

This paper constructs a specific purely unrectifiable measure in Euclidean space where certain singular integrals with harmonic polynomial kernels are bounded in L^2, contrasting with known results for Riesz kernels.

Contribution

It provides a novel example of unrectifiable measure with bounded singular integrals for kernels involving harmonic polynomials, challenging previous assumptions.

Findings

Constructed a purely unrectifiable measure with bounded singular integrals.

Demonstrated contrast with Riesz kernel behavior.

Showed boundedness for kernels with harmonic polynomial factors.

Abstract

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