On general local Tb theorems

TL;DR

This paper advances the theory of local Tb theorems by establishing results for upper doubling measures and refining proof techniques for doubling measures, broadening the scope of applicable measure classes.

Contribution

It introduces a local Tb theorem for upper doubling measures and adapts existing methods to prove results with L^2 testing conditions for doubling measures.

Findings

Proved a local Tb theorem for upper doubling measures.

Modified proof techniques for the doubling case.

Established scale-invariant testing conditions in general settings.

Abstract

In this paper, local Tb theorems are studied both in the doubling and non-doubling situation. We prove a local Tb theorem for the class of upper doubling measures. With such general measures, scale invariant testing conditions are required (L^{\infty} or BMO). In the case of doubling measures, we also modify the general non-homogeneous method of proof to yield a new proof of the local Tb theorem with L^2 type testing conditions.