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THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS

Part of: Stochastic processes Harmonic analysis in several variables Linear function spaces and their duals

Published online by Cambridge University Press:  26 August 2014

TUOMAS P. HYTÖNEN  and
ANTTI V. VÄHÄKANGAS
TUOMAS P. HYTÖNEN
Affiliation:
Department of Mathematics and Statistics, Gustaf Hällströmin katu 2b, FI-00014 University of Helsinki, Finland e-mail: tuomas.hytonen@helsinki.fi; antti.vahakangas@helsinki.fi
ANTTI V. VÄHÄKANGAS
Affiliation:
Department of Mathematics and Statistics, Gustaf Hällströmin katu 2b, FI-00014 University of Helsinki, Finland e-mail: tuomas.hytonen@helsinki.fi; antti.vahakangas@helsinki.fi
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Abstract

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We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) property’. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 46E40: Spaces of vector- and operator-valued functions 60G46: Martingales and classical analysis

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 1 , January 2015 , pp. 17 - 82
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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