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A $ T(1)$-theorem in relation to a semigroup of operators and applications to new paraproducts


Author: Frédéric Bernicot
Journal: Trans. Amer. Math. Soc. 364 (2012), 6071-6108
MSC (2010): Primary 30E20, 42B20, 42B30
DOI: https://doi.org/10.1090/S0002-9947-2012-05609-1
Published electronically: April 30, 2012
MathSciNet review: 2946943
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Abstract: In this work, we are interested in developing new directions of the famous $ T(1)$-theorem. More precisely, we develop a general framework where we look to replace the John-Nirenberg space $ BMO$ (in the classical result) by a new $ BMO_{L}$, associated to a semigroup of operators $ (e^{-tL})_{t>0}$. These new spaces $ BMO_L$ (including $ BMO$) have recently appeared in numerous works in order to extend the theory of Hardy and $ BMO$ space to more general situations. Then we give applications by describing boundedness for a new kind of paraproduct, built on the considered semigroup. In addition we obtain a version of the classical $ T(1)$-theorem for doubling Riemannian manifolds.


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Additional Information

Frédéric Bernicot
Affiliation: CNRS - Laboratoire Paul Painlevé, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France
Address at time of publication: CNRS - Laboratoire Jean Leray, Université de Nantes, 2, rue de la Houssinière, 44322 Nantes, cedex 3, France
Email: frederic.bernicot@math.univ-lille1.fr

DOI: https://doi.org/10.1090/S0002-9947-2012-05609-1
Keywords: $T(1)$-theorem, semigroup of operators, paraproducts
Received by editor(s): April 22, 2010
Received by editor(s) in revised form: April 25, 2011
Published electronically: April 30, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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