Archiv der Mathematik

, Volume 113, Issue 1, pp 81–93 | Cite as

Fefferman–Stein inequalities for the dyadic-like maximal operators

  • Mateusz RapickiEmail author
Open Access


The paper contains a transference theorem which allows to extend a large class of unweighted inequalities for the dyadic maximal operator to their weighted Fefferman–Stein counterparts on general probability spaces.


Maximal operator Weighted inequality Fefferman–Stein inequality Martingale Bellman function method 

Mathematics Subject Classification

Primary 42B25 Secondary 60G42 



Funding was provided by Narodowe Centrum Nauki (Grant No. DEC-2014/14/E/ST1/00532).


  1. 1.
    Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)zbMATHGoogle Scholar
  2. 2.
    Burkholder, D.L.: Explorations in Martingale Theory and Its Applications. École d’Été de Probabilités de Saint-Flour XIX–1989, pp. 1–66, Lecture Notes in Mathematics, vol. 1464. Springer, Berlin (1991)Google Scholar
  3. 3.
    Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gilat, D.: The best bound in the LlogL inequality of Hardy and Littlewood and its martingale counterpart. Proc. Am. Math. Soc. 97(3), 429–436 (1986)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6, 777–818 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Melas, A.D.: The Bellman functions of dyadic-like maximal operators and related inequalities. Adv. Math. 192, 310–340 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Melas, A.D., Nikolidakis, E.: On weak type inequalities for dyadic maximal functions. J. Math. Anal. Appl. 348, 404–410 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Nazarov, F., Treil, S.: The hunt for Bellman function: applications to estimates of singular integral operators and to other classical problems in harmonic analysis. Algebra Anal. 8, 32–162 (1997)zbMATHGoogle Scholar
  9. 9.
    Osękowski, A.: Sharp Martingale and Semimartingale Inequalities. Monografie Matematyczne, vol. 72, Birkhäuser Basel (2012)Google Scholar
  10. 10.
    Osękowski, A.: Sharp weak-type estimates for the dyadic-like maximal operators. Taiwan. J. Math. 19, 1031–1050 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Osękowski, A.: Sharp weighted logarithmic bound for maximal operators. Arch. Math. 107, 635–644 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Osękowski, A.: Sharp weighted bounds for geometric maximal operators. Glasg. Math. J. 59, 533–547 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Osękowski, A.: Sharp \(L_p\)-bounds for the martingale maximal function. Tohoku Math. J. 70, 121–138 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vasyunin, V., Volberg, A.: Monge–Ampère equation and Bellman optimization of Carleson embedding theorems. In: Linear Operators and Complex Analysis, pp. 195–238. American Mathematical Society, Providence (2009)Google Scholar

Copyright information

© The Author(s) 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Wydzial Matematyki Informatyki i MechanikiUniwersytet WarszawskiWarsawPoland

Personalised recommendations