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On the Local Tb Theorem: A Direct Proof under the Duality Assumption

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  13 February 2015

Michael T. Lacey  and
Antti V. Vähäkangas
Michael T. Lacey
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (lacey@math.gatech.edu)
Antti V. Vähäkangas
Affiliation:
Department of Mathematics and Statistics, PO Box 68, 00014, University of Helsinki, Finland (antti.vahakangas@helsinki.fi)
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Abstract

We give a new direct proof of the local Tb theorem in the Euclidean setting and under the assumption of dual exponents. This theorem provides a flexible framework for proving the boundedness of a Calderón–Zygmund operator, supposing the existence of systems of local accretive functions. We assume that the integrability exponents on these systems of functions are of the form 1/p + 1/q ⩽ 1, the ‘dual case’ 1/p + 1/q = 1 being the most difficult one. Our proof is direct: it avoids a reduction to the perfect dyadic case unlike some previous approaches. The principal point of interest is in the use of random grids and the corresponding construction of the corona. We also use certain twisted martingale transform inequalities.

Keywords

local Tb theoremTl theoremcoronatwisted martingale transformstopping cubes

MSC classification

Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 193 - 222
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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