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

Published online by Cambridge University Press:  13 February 2015

Michael T. Lacey
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA ([email protected])
Antti V. Vähäkangas
Affiliation:
Department of Mathematics and Statistics, PO Box 68, 00014, University of Helsinki, Finland ([email protected])

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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