Singular integrals unsuitable for the curvature method whose $L^2$-boundedness still implies rectifiability

TL;DR

This paper presents a novel example of a singular integral operator in the plane where the permutations change sign, yet its $L^2$-boundedness still implies the rectifiability of the measure's support, challenging existing curvature method limitations.

Contribution

It introduces the first known example of an operator with sign-changing permutations that still links $L^2$-boundedness to rectifiability, expanding the applicability of geometric measure theory.

Findings

An example of a singular integral operator with sign-changing permutations.

$L^2$-boundedness implies rectifiability despite sign changes.

Results hold under Ahlfors-David regularity conditions.

Abstract

The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the $L^2(\mu)$-boundedness of certain singular integral operators to the geometric properties of the support of measure $\mu$, e.g. rectifiability. It can be applied however only if Menger curvature-like permutations, directly associated with the kernel of the operator, are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the $L^2(\mu)$-boundedness of the operator still implies that the support of $\mu$ is rectifiable. To the best of our knowledge, it is the first example of this type. We also obtain several related results with Ahlfors-David regularity conditions.