The Perfect Local $ Tb$ Theorem and Twisted Martingale Transforms

TL;DR

This paper presents a new, simplified proof of the local Tb Theorem for perfect dyadic models of Calderón-Zygmund operators, utilizing twisted martingale transforms and minimal integrability conditions.

Contribution

It introduces a direct proof approach for the local Tb Theorem in dyadic models, employing twisted martingale transforms and optimal integrability assumptions.

Findings

New inequality for twisted martingale transforms

Simplified proof of the local Tb Theorem

Weakest integrability conditions on functions b_Q

Abstract

A local Tb Theorem provides a flexible framework for proving the boundedness of a Calder\'on-Zygmund operator T. One needs only boundedness of the operator T on systems of locally pseudo-accretive functions \{b_Q\}, indexed by cubes. We give a new proof of this Theorem in the setting of perfect (dyadic) models of Calder\'on-Zygmund operators, imposing integrability conditions on the b_Q functions that are the weakest possible. The proof is a simple direct argument, based upon a new inequality for transforms of so-called twisted martingale differences.