Dyadic harmonic analysis beyond doubling measures

TL;DR

This paper characterizes measures on the real line for which dyadic Hilbert transforms are bounded, revealing a class larger than doubling measures, and extends the analysis to higher dimensions with a new Calderón-Zygmund decomposition.

Contribution

It introduces a new Calderón-Zygmund decomposition for arbitrary Borel measures and characterizes boundedness of dyadic Hilbert transforms beyond traditional doubling measures.

Findings

Class of measures for bounded dyadic Hilbert transform is larger than doubling measures.

Complete characterization of weak-type (1,1) for Haar shift operators in higher dimensions.

Development of a new Calderón-Zygmund decomposition applicable to all Borel measures.

Abstract

We characterize the Borel measures $\mu$ on $\mathbb{R}$ for which the associated dyadic Hilbert transform, or its adjoint, is of weak-type $(1,1)$ and/or strong-type $(p,p)$ with respect to $\mu$. Surprisingly, the class of such measures is strictly bigger than the traditional class of dyadically doubling measures and strictly smaller than the whole Borel class. In higher dimensions, we provide a complete characterization of the weak-type $(1,1)$ for arbitrary Haar shift operators, cancellative or not, written in terms of two generalized Haar systems and these include the dyadic paraproducts. Our main tool is a new Calder\'on-Zygmund decomposition valid for arbitrary Borel measures which is of independent interest.