Three revolutions in the kernel are worse than one

TL;DR

This paper constructs a measure demonstrating that three kernel transformations can lead to bounded singular integrals, contrasting with known results for the Cauchy transform, highlighting complex behaviors in harmonic analysis.

Contribution

It introduces a novel measure showing that multiple kernel transformations can produce bounded singular integrals, challenging existing understanding in harmonic analysis.

Findings

Constructed a purely unrectifiable measure with bounded singular integral

Demonstrated that three kernel transformations can be bounded

Contrasts with known unboundedness of the Cauchy transform

Abstract

An example is constructed of a purely unrectifiable measure $\mu$ for which the singular integral operator whose kernel triples and reverses the argument of a complex number is bounded $L^2(\mu)$. This is in sharp contrast with the results known for the Cauchy transform, whose kernel reverses the argument of a complex number.