Local Tb theorem with L2 testing conditions and general measures: Square functions

TL;DR

This paper proves a new non-homogeneous local Tb theorem with L2 testing conditions for square functions using general measures, overcoming previous scale invariance limitations.

Contribution

It introduces the first non-homogeneous local Tb theorem with L2 testing conditions in the setting of general measures and kernels.

Findings

Established a non-homogeneous local Tb theorem with L2 testing conditions.

Extended the theory to general measures and kernels for square functions.

Utilized innovative techniques like Whitney averaging and Calderon-Zygmund stopping data.

Abstract

Local Tb theorems with Lp type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. In the non-homogeneous world local Tb theorems have only been proved assuming scale invariant ($L^{\infty}$ or BMO) testing conditions. In this paper, for the first time, we overcome these obstacles in the non-homogeneous world, and prove a non-homogeneous local Tb theorem with L2 type testing conditions. This paper is in the setting of the vertical and conical square functions defined using general measures and kernels. The proof uses various recent innovations including a Whitney averaging formula and the insertion of a Calderon-Zygmund stopping data of a fixed function in to the construction of the twisted martingale difference operators.