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Weak Type Estimates for the Noncommutative Vilenkin–Fourier Series

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Abstract

Let \({\mathcal {R}}\) be the separable hyperfinite factor of type \(\text {II}_1\). We show that for any bounded Vilenkin group, the sequence of partial sums of the corresponding noncommutative Vilenkin–Fourier series is a uniformly bounded family of weak type (1, 1) operators.

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References

  1. Agaev, G.N., Ya, N., Vilenkin, Dzhafarli, G.M., Rubinshteĭn, A.I.: Мультипликативные  системы  функциĭ  и  гармоническиĭ  анализ  на  нульмерных  группах “Èlm”, Baku (1981)

  2. Bennett, C., Sharpley, R.: Interpolation of Operators, Volume 129 of Pure and Applied Mathematics. Academic Press, Inc, Boston (1988). ISBN: 0-12-088730-4

    MATH  Google Scholar 

  3. Cuculescu, I.: Martingales on von Neumann algebras. J. Multivar. Anal. 1(1), 17–27 (1971). https://doi.org/10.1016/0047-259X(71)90027-3

    Article  MathSciNet  MATH  Google Scholar 

  4. Dodds, P.G., Sukochev, F.A.: Non-commutative bounded Vilenkin systems. Math. Scand. 87(1), 73–92 (2000). https://doi.org/10.7146/math.scand.a-14300

    Article  MathSciNet  MATH  Google Scholar 

  5. Dodds, P.G., Dodds, T.K.-Y., de Pagter, B.: Noncommutative Banach function spaces. Math. Z. 201(4), 583–597 (1989). https://doi.org/10.1007/BF01215160

    Article  MathSciNet  MATH  Google Scholar 

  6. Dodds, P.G., Ferleger, S.V., de Pagter, B., Sukochev, F.A.: Vilenkin systems and generalized triangular truncation operator. Integral Equ. Oper. Theory 40(4), 403–435 (2001). https://doi.org/10.1007/BF01198137

    Article  MathSciNet  MATH  Google Scholar 

  7. Fack, T., Kosaki, H.: Generalized \(s\)-numbers of \(\tau \)-measurable operators. Pac. J. Math. 123(2), 269–300 (1986). ISSN 0030-8730. http://projecteuclid.org/euclid.pjm/1102701004

  8. Golubov, B., Efimov, A., Skvortsov, V.: Walsh Series and Transforms, Volume 64 of Mathematics and Its Applications (Soviet Series). Theory and Applications, Translated from the 1987 Russian original by W. R. Wade. Kluwer Academic Publishers Group, Dordrecht (1991). https://doi.org/10.1007/978-94-011-3288-6, ISBN: 0-7923-1100-0

  9. Gosselin, J.: Almost everywhere convergence of Vilenkin–Fourier series. Trans. Am. Math. Soc. 185(345–370), 1973 (1974). https://doi.org/10.2307/1996444

    Article  MathSciNet  MATH  Google Scholar 

  10. Grafakos, L.: Classical Fourier Analysis, Volume 249 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (2008). ISBN: 978-0-387-09431-1

    Google Scholar 

  11. Jiao, Y., Zhou, D., Wu, L., Zanin, D.: Noncommutative dyadic martingales and Walsh–Fourier series. J. Lond. Math. Soc. (2) 97(3), 550–574 (2018)

    Article  MathSciNet  Google Scholar 

  12. Junge, M., Xu, Q.: On the best constants in some non-commutative martingale inequalities. Bull. Lond. Math. Soc. 37(2), 243–253 (2005). https://doi.org/10.1112/S0024609304003947

    Article  MathSciNet  MATH  Google Scholar 

  13. Lord, S., Sukochev, F.A., Zanin, D.: Singular Traces, Volume 46 of De Gruyter Studies in Mathematics. De Gruyter, Berlin (2013). ISBN: 978-3-11-026250-6; 978-3-11-026255-1. (Theory and applications)

  14. Lust-Piquard, F., Pisier, G.: Noncommutative Khintchine and Paley inequalities. Ark. Mat. 29(2), 241–260 (1991). https://doi.org/10.1007/BF02384340

    Article  MathSciNet  MATH  Google Scholar 

  15. Parcet, J.: Pseudo-localization of singular integrals and noncommutative Calderón–Zygmund theory. J. Funct. Anal. 256(2), 509–593 (2009). https://doi.org/10.1016/j.jfa.2008.04.007

    Article  MathSciNet  MATH  Google Scholar 

  16. Parcet, J., Randrianantoanina, N.: Gundy’s decomposition for non-commutative martingales and applications. Proc. Lond. Math. Soc. (3) 93(1), 227–252 (2006). https://doi.org/10.1017/S0024611506015863

    Article  MathSciNet  MATH  Google Scholar 

  17. Pisier, G.: Martingales in Banach Spaces, Volume 155 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016). ISBN: 978-1-107-13724-0

    Google Scholar 

  18. Pisier, G., Xu, Q.: Non-commutative martingale inequalities. Commun. Math. Phys. 189(3), 667–698 (1997). https://doi.org/10.1007/s002200050224

    Article  MathSciNet  MATH  Google Scholar 

  19. Pisier, G., Xu, Q.: Non-commutative \(L^p\)-spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517. North-Holland, Amsterdam (2003). https://doi.org/10.1016/S1874-5849(03)80041-4

    Chapter  MATH  Google Scholar 

  20. Randrianantoanina, N.: Non-commutative martingale transforms. J. Funct. Anal. 194(1), 181–212 (2002). https://doi.org/10.1006/jfan.2002.3952

    Article  MathSciNet  MATH  Google Scholar 

  21. Randrianantoanina, N.: A weak type inequality for non-commutative martingales and applications. Proc. Lond. Math. Soc. (3) 91(2), 509–542 (2005). https://doi.org/10.1112/S0024611505015297

    Article  MathSciNet  MATH  Google Scholar 

  22. Schipp, F., Wade, W.R., Simon, P.: Walsh Series. Adam Hilger, Ltd., Bristol (1990). ISBN: 0-7503-0068-X. An introduction to dyadic harmonic analysis, with the collaboration of J. Pál

  23. Sukochev, F.A., Zanin, D.: Johnson–Schechtman inequalities in the free probability theory. J. Funct. Anal. 263(10), 2921–2948 (2012). https://doi.org/10.1016/j.jfa.2012.07.012

    Article  MathSciNet  MATH  Google Scholar 

  24. Takesaki, M.: Theory of Operator Algebras. III, Volume 127 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2003). ISBN: 3-540-42913-1. https://doi.org/10.1007/978-3-662-10453-8. (Operator Algebras and Non-commutative Geometry, 8)

  25. Watari, C.: On generalized Walsh Fourier series. Tôhoku Math. J. 2(10), 211–241 (1958). https://doi.org/10.2748/tmj/1178244661

    Article  MathSciNet  MATH  Google Scholar 

  26. Young, W.-S.: Mean convergence of generalized Walsh–Fourier series. Trans. Am. Math. Soc. 218, 311–320 (1976). https://doi.org/10.2307/1997441

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, 2052, Australia

    Thomas Tzvi Scheckter & Fedor Sukochev

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  1. Thomas Tzvi Scheckter
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  2. Fedor Sukochev
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Correspondence to Thomas Tzvi Scheckter.

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The first named author was supported by an Australian Government Research Training Program (RTP) Scholarship.

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Scheckter, T.T., Sukochev, F. Weak Type Estimates for the Noncommutative Vilenkin–Fourier Series. Integr. Equ. Oper. Theory 90, 64 (2018). https://doi.org/10.1007/s00020-018-2489-8

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  • DOI: https://doi.org/10.1007/s00020-018-2489-8

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