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

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

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

References

REFERENCES

1.Bourgain, J., Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (2) (1983), 163–168.Google Scholar
2.Bourgain, J., Vector-valued singular integrals and the H 1-BMO duality, in Probability theory and harmonic analysis (Cleveland, Ohio, 1983), volume 98 of Monogr. Textbooks Pure Appl. Math. (Dekker, New York, 1986), 1–19.Google Scholar
3.Burkholder, D. L., A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, in Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA (1983, 270–286.Google Scholar
4.Hytönen, T. P., The vector-valued non-homogeneous Tb Theorem, Int. Math. Res. Not. (online, 2012) doi: 10.1093/imrn/rns222.Google Scholar
5.Hytönen, T. and Kemppainen, M., On the relation of Carleson's embedding and the maximal theorem in the context of Banach space geometry, Math. Scand. 109 (2), 269–284 (2011).CrossRefGoogle Scholar
6.Hytönen, T. and Martikainen, H., Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces, J. Geom. Anal. 22 (4) (2012), 1071–1107.Google Scholar
7.Hytönen, T., McIntosh, A. and Portal, P., Kato's square root problem in Banach spaces, J. Funct. Anal. 254 (3) (2008), 675–726.CrossRefGoogle Scholar
8.Kemppainen, M., On the Rademacher maximal function, Studia Math. 203 (1) (2011) 1–31.Google Scholar
9.Maurey, B., Système de Haar, in Séminaire Maurey-Schwartz 1974–1975: Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. I et II (erratum, p. 1). Centre Math. (École Polytech., Paris, 1975), 26.Google Scholar
10.Mayboroda, S. and Volberg, A., Boundedness of the square function and rectifiability, C. R. Math. Acad. Sci. Paris 347 (17–18) (2009), 1051–1056.CrossRefGoogle Scholar
11.McConnell, T. R., Decoupling and stochastic integration in UMD Banach spaces, Probab. Math. Statist. 10 (2) (1989), 283–295.Google Scholar
12.Nazarov, F., Treil, S. and Volberg, A., The Tb-theorem on non-homogeneous spaces, Acta Math. 190 (2) (2003), 151–239.Google Scholar
13.Nazarov, F., Treil, S. and Volberg, A., Accretive system Tb-theorems on nonhomogeneous spaces, Duke Math. J. 113 (2) (2002) 259–312.Google Scholar
14.Rubio de Francia, José L., Martingale and integral transforms of Banach space valued functions, in Probability and Banach spaces (Zaragoza, 1985), volume 1221 of Lecture Notes in Math., (Springer, Berlin, 1986) 195–222.CrossRefGoogle Scholar
15.Elias, M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Mathematics Studies, No. 63 (Princeton University Press, Princeton, N.J., 1970).Google Scholar
16.Volberg, A., The proof of non-homogeneous T1 theorem via averaging of dyadic shifts. Preprint, arXiv:1303.0367, 2013.Google Scholar
17.Weis, L., Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Math. Ann. 319 (4) (2001) 735–758.Google Scholar