Geodesic
THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS
Glasgow mathematical journal, Tome 57 (2015) no. 1, pp. 17-82

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
HYTÖNEN, TUOMAS P.; VÄHÄKANGAS, ANTTI V. THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS. Glasgow mathematical journal, Tome 57 (2015) no. 1, pp. 17-82. doi: 10.1017/S0017089514000123
@article{10_1017_S0017089514000123,
     author = {HYT\"ONEN, TUOMAS P. and V\"AH\"AKANGAS, ANTTI V.},
     title = {THE {LOCAL} {NON-HOMOGENEOUS} {Tb} {THEOREM} {FOR} {VECTOR-VALUED} {FUNCTIONS}},
     journal = {Glasgow mathematical journal},
     pages = {17--82},
     year = {2015},
     volume = {57},
     number = {1},
     doi = {10.1017/S0017089514000123},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000123/}
}
TY  - JOUR
AU  - HYTÖNEN, TUOMAS P.
AU  - VÄHÄKANGAS, ANTTI V.
TI  - THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS
JO  - Glasgow mathematical journal
PY  - 2015
SP  - 17
EP  - 82
VL  - 57
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000123/
DO  - 10.1017/S0017089514000123
ID  - 10_1017_S0017089514000123
ER  - 
%0 Journal Article
%A HYTÖNEN, TUOMAS P.
%A VÄHÄKANGAS, ANTTI V.
%T THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS
%J Glasgow mathematical journal
%D 2015
%P 17-82
%V 57
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000123/
%R 10.1017/S0017089514000123
%F 10_1017_S0017089514000123

[1] 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] 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] 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] 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] 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). Google Scholar | DOI

[6] 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] 7.Hytönen, T., Mcintosh, A. and Portal, P., Kato's square root problem in Banach spaces, J. Funct. Anal. 254 (3) (2008), 675–726. Google Scholar | DOI

[8] 8.Kemppainen, M., On the Rademacher maximal function, Studia Math. 203 (1) (2011) 1–31. Google Scholar

[9] 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] 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. Google Scholar | DOI

[11] 11.Mcconnell, T. R., Decoupling and stochastic integration in UMD Banach spaces, Probab. Math. Statist. 10 (2) (1989), 283–295. Google Scholar

[12] 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] 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] 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. Google Scholar | DOI

[15] 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] 16.Volberg, A., The proof of non-homogeneous T1 theorem via averaging of dyadic shifts. Preprint, arXiv:1303.0367, 2013. Google Scholar

[17] 17.Weis, L., Operator-valued Fourier multiplier theorems and maximal L-regularity, Math. Ann. 319 (4) (2001) 735–758. Google Scholar

Cité par Sources :