A new approach to non-homogeneous local $Tb$ theorems

TL;DR

This paper introduces a novel method for establishing non-homogeneous local Tb theorems, integrating existing techniques and proposing new measure change strategies, leading to improved results and broader applications.

Contribution

It develops a new general approach that combines multiple existing methods with innovative measure change techniques for non-doubling local Tb theorems.

Findings

Improved non-homogeneous local Tb theorems.

Integration of Tolsa's good lambda method with Nazarov-Treil-Volberg's big pieces theorem.

New applications and extensions of existing results.

Abstract

We develop a new general method to prove various non-doubling local Tb theorems. The method combines the non-homogeneous good lambda method of Tolsa, the big pieces Tb theorem of Nazarov-Treil-Volberg and a new change of measure argument based on stopping time techniques. We also improve known results and discuss some further applications.