A weak to strong type T1 theorem for general smooth Calder\'on-Zygmund operators with doubling weights, II

TL;DR

This paper establishes a comprehensive criterion linking weak and strong type inequalities for Calderón-Zygmund operators with doubling weights, extending to a weak Tb theorem and relating maximal truncations.

Contribution

It proves that weak (2,2) boundedness implies strong (2,2) boundedness under dual testing conditions, extending the theory to general smooth Calderón-Zygmund operators with doubling weights.

Findings

Weak to strong type equivalence under dual testing conditions.

Extension of results to a weak form of Tb theorem.

Equivalence of weak type inequalities for operators and their maximal truncations.

Abstract

We consider the weak to strong type problem for two weight norm inequalities for Calder\'on-Zygmund operators with doubling weights. We show that if a Calder\'on-Zygmund operator T is weak type (2,2) with doubling weights, then it is strong type (2,2) if and only if the dual cube testing condition for T^{*} holds, alternatively if and only if the dual cancellation condition of Stein holds. The testing condition can be taken with respect to either cubes or balls, and more generally, this is extended to a weak form of Tb theorem. Finally, we show that for all pairs of locally finite positive Borel measures, and all Stein elliptic Calder\'on-Zygmund operators T, the weak type (2,2) inequalities for T and and its associated maximal truncations operator T_{*} are equivalent. Thus the characterization of weak type for T_{*} in [LaSaUr1] applies to T as well.