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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On general local $ Tb$ theorems


Authors: Tuomas Hytönen and Henri Martikainen
Journal: Trans. Amer. Math. Soc. 364 (2012), 4819-4846
MSC (2010): Primary 42B20; Secondary 42B25
Published electronically: April 16, 2012
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Abstract: In this paper, local $ Tb$ theorems are studied both in the doubling and non-doubling situation. We prove a local $ Tb$ theorem for the class of upper doubling measures. With such general measures, scale invariant testing conditions are required ( $ L^{\infty }$ or BMO). In the case of doubling measures, we also modify the general non-homogeneous method of proof to yield a new proof of the local $ Tb$ theorem with $ L^2$ type testing conditions.


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Additional Information

Tuomas Hytönen
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O.B. 68, FI-00014 Helsinki, Finland
Email: tuomas.hytonen@helsinki.fi

Henri Martikainen
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O.B. 68, FI-00014 Helsinki, Finland
Email: henri.martikainen@helsinki.fi

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05599-1
PII: S 0002-9947(2012)05599-1
Keywords: Calderón–Zygmund operator, non-homogeneous theory, square function
Received by editor(s): November 4, 2010
Published electronically: April 16, 2012
Additional Notes: The authors were supported by the Academy of Finland through the project “$L^{p}$ methods in harmonic analysis”.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.