On the Local $ Tb$ Theorem under Minimal Integrability

TL;DR

This paper extends the local Tb Theorem to cases where the functions involved are only locally L^p integrable, broadening the theorem's applicability with new techniques like twisted martingale differences.

Contribution

It generalizes the local Tb Theorem by weakening integrability assumptions and introduces novel methods such as twisted martingale differences and random dyadic grids.

Findings

Proves the local Tb Theorem under minimal integrability conditions.

Introduces techniques involving twisted martingale differences.

Utilizes random dyadic grids in the proof.

Abstract

We prove a version of the local Tb Theorem assuming that the accretive functions b_Q and T b_Q are locally L ^{p} integrable, for any 1< p < \infty . This improves a recent result of Hytonen-Nazarov. The proof strategy relies upon the their strategy, with additional techniques concerning twisted martingale differences and the use of random dyadic grids in the local Tb setting.