Inventiones mathematicae

, Volume 209, Issue 3, pp 665–748 | Cite as

\(\ell ^p\left( \mathbb {Z}^d\right) \)-estimates for discrete operators of Radon type: variational estimates

  • Mariusz Mirek
  • Elias M. Stein
  • Bartosz Trojan


We prove \(\ell ^p\left( {\mathbb {Z}}^d\right) \) bounds for \(p\in (1, \infty )\), of r-variations \(r\in (2, \infty )\), for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of p and r which coincide with the ranges of their continuous counterparts.


  1. 1.
    Bourgain, J.: Pointwise ergodic theorems for arithmetic sets. With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson and Donald S. Ornstein. Publ. Math. Paris 69(1), 5–45 (1989)CrossRefzbMATHGoogle Scholar
  2. 2.
    Carbery, A., Christ, M., Wright, J.: Multidimensional van der Corput and sublevel set estimates. J. Am. Math. Soc. 12(4), 981–1015 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Christ, M.: A \({T}(b)\) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 2(60–61), 601–628 (1990)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cotlar, M.: A unified theory of Hilbert transforms and ergodic theorems. Rev. Mat. Cuyana 1(2), 105–167 (1955)MathSciNetGoogle Scholar
  5. 5.
    deLeeuw, K.: On \(L^p\) multipliers. Ann. Math. 81, 364–379 (1965)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84(3), 541–561 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hughes, K., Krause, B., Trojan, B.: The maximal function and conditional square function control the variation: an elementary proof. Proc. Amer. Math. Soc. 144(8), 3583–3588 (2016)Google Scholar
  8. 8.
    Ionescu, A.D., Wainger, S.: \(L^p\) boundedness of discrete singular Radon transforms. J. Am. Math. Soc. 19(2), 357–383 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jones, R.L., Kaufman, R., Rosenblatt, J.M., Wierdl, M.: Oscillation in ergodic theory. Ergod. Theory Dyn. Syst. 18(4), 889–935 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jones, R.L., Seeger, A., Wright, J.: Strong variational and jump inequalities in harmonic analysis. Trans. Am. Math. Soc. 6711–6742 (2008)Google Scholar
  11. 11.
    Krause, B.: Polynomial ergodic averages converge rapidly: variations on a theorem of Bourgain. arXiv:1402.1803 (2014)
  12. 12.
    Krause, B.: Some optimizations for (maximal) multipliers in \({L}^{p}\). arXiv:1402.1804 (2014)
  13. 13.
    Lepingle, D.: La variation d’ordre \(p\) des semi-martingales. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 36(4), 295–316 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Magyar, A., Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis: spherical averages. Ann. Math. 189–208 (2002)Google Scholar
  15. 15.
    Mirek, M., Stein, E.M., Trojan, B.: \({\ell }^{p}\left({\mathbb{Z}}^d\right)\)-estimates for discrete operators of Radon type: Maximal functions and vector-valued estimates. arXiv:1512.07518 (2015)
  16. 16.
    Mirek, M., Trojan, B.: Discrete maximal functions in higher dimensions and applications to ergodic theory. Am. J. Math. 138(6), 1495–1532 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nazarov, F., Oberlin, R., Thiele, Ch.: A Calderón Zygmund decomposition for multiple frequencies and an application to an extension of a lemma of Bourgain. Math. Res. Lett. 17(3), 529–545 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pisier, G., Xu, Q.: The strong \(p\)-variation of martingales and orthogonal series. Probab. Theory Relat. Fields 77(4), 497–514 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  20. 20.
    Stein, E.M., Wainger, S.: Discrete analogues in harmonic analysis, I: \(\ell ^2\) estimates for singular Radon transforms. Am. J. Math. 121(6), 1291–1336 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Stein, E.M., Wainger, S.: Oscillatory integrals related to Carleson’s theorem. Math. Res. Lett. 8, 789–800 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zorin-Kranich, P.: Variation estimates for averages along primes and polynomials. J. Funct. Anal. 268(1), 210–238 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Mariusz Mirek
    • 1
    • 2
  • Elias M. Stein
    • 3
  • Bartosz Trojan
    • 4
  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA
  4. 4.Wrocław University of TechnologyWrocławPoland

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