Singular integrals with variable kernels in dyadic settings

TL;DR

This paper investigates Calderón-Zygmund operators with variable kernels in dyadic settings, extending Petermichl's dyadic kernel to spaces of homogeneous type and analyzing conditions on variable symbols relative to Haar systems.

Contribution

It introduces a framework for variable kernel singular integrals in dyadic spaces and generalizes Petermichl's kernel to spaces of homogeneous type.

Findings

Petermichl's dyadic kernel is a variable kernel singular integral.

Extended dyadic systems to spaces of homogeneous type.

Established conditions on variable symbols with respect to Haar systems.

Abstract

In this paper we explore conditions on variable symbols with respect to Haar systems, defining Calder\'on-Zygmund type operators with respect to the dyadic metrics associated to the Haar bases.We show that Petermichl's dyadic kernel can be seen as a variable kernel singular integral and we extend it to dyadic systems built on spaces of homogeneous type.