A Good-Lambda Lemma, two weight T1 theorems without weak boundedness, and a two weight accretive global Tb theorem

TL;DR

This paper establishes new two-weight boundedness criteria for Calderon-Zygmund operators, removing the weak boundedness property via good-lambda control and deriving a two-weight accretive global Tb theorem, advancing the theory of weighted inequalities.

Contribution

It introduces a good-lambda lemma that controls the weak boundedness property and removes it from key theorems, providing new two-weight T1 and Tb theorems without weak boundedness assumptions.

Findings

Weak boundedness property is controlled by testing and side conditions.

Boundedness of T is characterized by testing conditions alone.

Derived a simple two-weight accretive global Tb theorem.

Abstract

The weak boundedness property associated with a standard alpha-fractional Calderon-Zygmund operator and a weight pair is good-lambda controlled by the testing conditions and the Muckenhoupt and energy side conditions. As a consequence, assuming the side conditions, we can eliminate the weak boundedness property from Theorem 1 of arXiv:1603.04332v2 to obtain that T is bounded if and only if the testing conditions hold for T and its dual (an earlier instance of this type of conclusion appears in Lacey and Wick arXiv:1312.6163v3). As a corollary we give a simple derivation of a two weight accretive global Tb theorem from a related T1 theorem. The role of two different parameterizations of the family of dyadic grids, by scale and by translation, is highlighted in simultaneously exploiting both goodness and NTV surgery with families of grids that are common to both measures.