On the Local $Tb$ Theorem: A Direct Proof under Duality Assumption

TL;DR

This paper presents a direct proof of the local Tb Theorem in Euclidean spaces, leveraging duality assumptions, random grids, and twisted martingale inequalities to establish boundedness of Calderón-Zygmund operators.

Contribution

It offers a novel direct proof of the local Tb Theorem under dual exponents, simplifying previous approaches and expanding the framework for Calderón-Zygmund operator analysis.

Findings

Proof of the local Tb Theorem under duality assumption.

Use of random grids and corona construction in the proof.

Application of twisted martingale transform inequalities.

Abstract

We give a direct proof of the local $Tb$ Theorem, in the Euclidean setting, and under the assumption of dual exponents. This Theorem provides a flexible framework for proving the boundedness of a Calder\'on-Zygmund operator, supposing the existence of systems of local accretive functions. We assume that the integrability exponents on these systems of functions are of the form $1/p+1/q\le 1$, and provide a direct proof. The principal point of interest is in the use of random grids and the corresponding construction of the corona. We also utilize certain twisted martingale transform inequalities.