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The perfect local $ Tb$ Theorem and twisted Martingale transforms


Authors: Michael T. Lacey and Antti V. Vähäkangas
Journal: Proc. Amer. Math. Soc. 142 (2014), 1689-1700
MSC (2010): Primary 42B20
DOI: https://doi.org/10.1090/S0002-9939-2014-11930-7
Published electronically: February 17, 2014
MathSciNet review: 3168475
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Abstract: A local $ Tb$ Theorem provides a flexible framework for proving the boundedness of a Calderón-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 Calderón-Zygmund operators, imposing integrability conditions on the $ b_Q$ functions that are the weakest possible. The proof is a simple direct argument, based upon an inequality for transforms of so-called twisted martingale differences, which has been noted by Auscher-Routin.


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Additional Information

Michael T. Lacey
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: lacey@math.gatech.edu

Antti V. Vähäkangas
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: antti.vahakangas@helsinki.fi

DOI: https://doi.org/10.1090/S0002-9939-2014-11930-7
Received by editor(s): April 29, 2012
Received by editor(s) in revised form: May 14, 2012, and June 19, 2012
Published electronically: February 17, 2014
Additional Notes: This research was supported in part by grant NSF-DMS 0968499 and a grant from the Simons Foundation (#229596) to the first author
The second author was supported by the School of Mathematics, Georgia Institute of Technology, and by the Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisälä Foundation
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.