T1 testing implies Tp polynomial testing: optimal cancellation conditions for CZO's

TL;DR

This paper extends the T1 theorem to more general weights, showing that standard testing conditions imply polynomial testing and establishing optimal cancellation conditions for Calderón-Zygmund operators.

Contribution

It demonstrates that classical T1 testing conditions combined with A2 conditions imply polynomial testing for general weights, advancing the theory of Calderón-Zygmund operators.

Findings

T1 testing conditions imply polynomial testing for CZO's.

Results apply to fractional singular integrals with doubling measures.

Provides a T1 theorem for fractional CZO's with optimal cancellation conditions.

Abstract

This paper is the third in an investigation begun in arXiv:1906.05602 and arXiv:1907.07571 of extending the T1 theorem of David and Journ\'e, and optimal cancellation conditions, to more general weight pairs. The main result here is that the familiar T1 testing conditions over indicators of cubes, together with the one-tailed A2 conditions, imply polynomial testing. Analogous results for fractional singular integrals hold as well. Applications include a T1 theorem for fractional CZO's T in the case of doubling measures when one of the weights is A infinity, and then to optimal cancellation conditions for such CZO's in similar situations.