Weak Type Estimates for the Noncommutative Vilenkin–Fourier Series

  • Thomas Tzvi ScheckterEmail author
  • Fedor Sukochev


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.


Noncommutative martingales Vilenkin groups Vilenkin–Fourier series 

Mathematics Subject Classification

Primary 46L52 Secondary 46L53 47A30 


  1. 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)Google Scholar
  2. 2.
    Bennett, C., Sharpley, R.: Interpolation of Operators, Volume 129 of Pure and Applied Mathematics. Academic Press, Inc, Boston (1988). ISBN: 0-12-088730-4zbMATHGoogle Scholar
  3. 3.
    Cuculescu, I.: Martingales on von Neumann algebras. J. Multivar. Anal. 1(1), 17–27 (1971). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dodds, P.G., Sukochev, F.A.: Non-commutative bounded Vilenkin systems. Math. Scand. 87(1), 73–92 (2000). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dodds, P.G., Dodds, T.K.-Y., de Pagter, B.: Noncommutative Banach function spaces. Math. Z. 201(4), 583–597 (1989). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fack, T., Kosaki, H.: Generalized \(s\)-numbers of \(\tau \)-measurable operators. Pac. J. Math. 123(2), 269–300 (1986). ISSN 0030-8730.
  8. 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)., ISBN: 0-7923-1100-0
  9. 9.
    Gosselin, J.: Almost everywhere convergence of Vilenkin–Fourier series. Trans. Am. Math. Soc. 185(345–370), 1973 (1974). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grafakos, L.: Classical Fourier Analysis, Volume 249 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (2008). ISBN: 978-0-387-09431-1Google Scholar
  11. 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)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Junge, M., Xu, Q.: On the best constants in some non-commutative martingale inequalities. Bull. Lond. Math. Soc. 37(2), 243–253 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)Google Scholar
  14. 14.
    Lust-Piquard, F., Pisier, G.: Noncommutative Khintchine and Paley inequalities. Ark. Mat. 29(2), 241–260 (1991). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Parcet, J.: Pseudo-localization of singular integrals and noncommutative Calderón–Zygmund theory. J. Funct. Anal. 256(2), 509–593 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Parcet, J., Randrianantoanina, N.: Gundy’s decomposition for non-commutative martingales and applications. Proc. Lond. Math. Soc. (3) 93(1), 227–252 (2006). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pisier, G.: Martingales in Banach Spaces, Volume 155 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016). ISBN: 978-1-107-13724-0Google Scholar
  18. 18.
    Pisier, G., Xu, Q.: Non-commutative martingale inequalities. Commun. Math. Phys. 189(3), 667–698 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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). CrossRefzbMATHGoogle Scholar
  20. 20.
    Randrianantoanina, N.: Non-commutative martingale transforms. J. Funct. Anal. 194(1), 181–212 (2002). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Randrianantoanina, N.: A weak type inequality for non-commutative martingales and applications. Proc. Lond. Math. Soc. (3) 91(2), 509–542 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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. PálGoogle Scholar
  23. 23.
    Sukochev, F.A., Zanin, D.: Johnson–Schechtman inequalities in the free probability theory. J. Funct. Anal. 263(10), 2921–2948 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Takesaki, M.: Theory of Operator Algebras. III, Volume 127 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2003). ISBN: 3-540-42913-1. (Operator Algebras and Non-commutative Geometry, 8)
  25. 25.
    Watari, C.: On generalized Walsh Fourier series. Tôhoku Math. J. 2(10), 211–241 (1958). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Young, W.-S.: Mean convergence of generalized Walsh–Fourier series. Trans. Am. Math. Soc. 218, 311–320 (1976). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesKensingtonAustralia

Personalised recommendations