A pointwise estimate for positive dyadic shifts and some applications

TL;DR

This paper establishes a linear-in-complexity pointwise estimate for positive dyadic shifts, enabling new bounds for Calderón-Zygmund operators and addressing a question by Lerner, with applications to weighted inequalities.

Contribution

It introduces a novel linear pointwise estimate for positive dyadic shifts of any complexity, improving understanding of their behavior and applications.

Findings

Derived a pointwise estimate for positive dyadic shifts of complexity m

Applied the estimate to Calderón-Zygmund operators and answered Lerner's question

Provided new weighted estimates for multilinear Calderón-Zygmund operators and square functions

Abstract

We prove a pointwise estimate for positive dyadic shifts of complexity $m$ which is linear in the complexity. This can be used to give a pointwise estimate for Calder\'on-Zygmund operators and to answer a question posed by A. Lerner. Several applications to weighted estimates for both multilinear Calder\'on-Zygmund operators and square functions are discussed.