Exotic Calder\'on-Zygmund operators

TL;DR

This paper introduces and analyzes a new class of singular integral operators with kernels more singular than classical Calderón-Zygmund kernels but less than bi-parameter product kernels, demonstrating their properties and limitations.

Contribution

It extends the theory of Calderón-Zygmund operators to a new class arising from Zygmund dilations, establishing T1 theorems and commutator estimates for these kernels.

Findings

Satisfy T1 theorem and commutator estimates similar to standard kernels

Fail to satisfy one-parameter weighted estimates in general

Connects these kernels to the broader framework of Calderón-Zygmund theory

Abstract

We study singular integral operators with kernels that are more singular than standard Calder\'on-Zygmund kernels, but less singular than bi-parameter product Calder\'on-Zygmund kernels. These kernels arise as restrictions to two dimensions of certain three-dimensional kernels adapted to so-called Zygmund dilations, which is part of our motivation for studying these objects. We make the case that such kernels can, in many ways, be seen as part of the extended realm of standard kernels by proving that they satisfy both a T1 theorem and commutator estimates in a form reminiscent of the corresponding results for standard Calder\'on-Zygmund kernels. However, we show that one-parameter weighted estimates, in general, fail.