The Perfect Local $ Tb$ Theorem and Twisted Martingale Transforms
Classical Analysis and ODEs
2015-09-02 v3
Abstract
A local Tb Theorem provides a flexible framework for proving the boundedness of a Calder\'on-Zygmund operator T. One needs only boundedness of the operator T on systems of locally pseudo-accretive functions \{b_Q\}, indexed by cubes. We give a new proof of this Theorem in the setting of perfect (dyadic) models of Calder\'on-Zygmund operators, imposing integrability conditions on the b_Q functions that are the weakest possible. The proof is a simple direct argument, based upon a new inequality for transforms of so-called twisted martingale differences.
Cite
@article{arxiv.1204.6526,
title = {The Perfect Local $ Tb$ Theorem and Twisted Martingale Transforms},
author = {Michael T. Lacey and Antti V. Vähäkangas},
journal= {arXiv preprint arXiv:1204.6526},
year = {2015}
}
Comments
12 pages v2: typos corrected in section 5 v3: The twisted martingale transform inequalities appeared in Auscher Routin (arXiv:1011.1747), which is now reflected in the text