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On the failure of lower square function estimates in the non-homogeneous weighted setting

  • K. Domelevo
  • P. Ivanisvili
  • S. Petermichl
  • S. Treil
  • A. Volberg
Article
  • 17 Downloads

Abstract

We show that the classical \(A_{\infty }\) condition is not sufficient for a lower square function estimate in the non-homogeneous weighted \(L^2\) space. We also show that under the martingale \(A_2\) condition, an estimate holds true, but the optimal power of the characteristic jumps from 1 / 2 to 1 even when considering the classical \(A_2\) characteristic. This is in a sharp contrast to known estimates in the dyadic homogeneous setting as well as the recent positive results in this direction on the discrete time non-homogeneous martingale transforms. Last, we give a sharp \(A_{\infty }\) estimate for the n-adic homogeneous case, growing with n.

References

  1. 1.
    Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics, vol. 129. Academic Press Inc., Boston, MA (1988)zbMATHGoogle Scholar
  2. 2.
    Bonami, A., Lépingle, D.: Fonction maximale et variation quadratique des martingales en présence d’un poids, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), Lecture Notes in Math., vol. 721, Springer, Berlin, 1979, pp. 294–306Google Scholar
  3. 3.
    Burkholder, D.L.: Martingale transforms. Ann. Math. Stat. 37, 1494–1504 (1966)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chang, S.-Y.A., Wilson, J.M., Wolff, T.H.: Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv. 60(2), 217–246 (1985)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hukovic, S., Treil, S., Volberg, A.: The Bellman functions and sharp weighted inequalities for square functions, Complex analysis, operators, and related topics. Oper. Theory Adv. Appl. vol. 113, Birkhäuser, Basel, pp. 97–113 (2000)Google Scholar
  6. 6.
    Hytonen, T., Perez, C.: Sharp weighted bounds involving \(A_{\infty }\). Anal. PDE 6(4), 777–818 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gundy, R.F., Wheeden, R.L.: Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh-Paley series. Stud. Math. 49 (1973/74), 107–124MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ivanisvili, P., Osipov, N.N., Stolyarov, D.M., Vasyunin, V.V., Zatitskiy, P.B.: Bellman function for extremal problems in BMO. Trans. Am. Math. Soc. 368, 3415–3468 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lacey, M.T.: An elementary proof of the \(A_2\) bound. Israel J. Math. 217(1), 181–195 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lacey, M.T., Li, K.: On \(A_p\)-\(A_\infty \) type estimates for square functions. Math. Z. 284(3–4), 1211–1222 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Nazarov, F., Petermichl, S., Treil, S., Volberg, A.: Convex body domination and weighted estimates with matrix weights. Adv. Math. 318, 279–306 (2017).  https://doi.org/10.1016/j.aim.2017.08.001 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Petermichl, S., Pott, S.: An estimate for weighted Hilbert transform via square functions. Trans. Am. Math. Soc. 354(4), 1699–1703 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Thiele, C., Treil, S., Volberg, A.: Weighted martingale multipliers in the non-homogeneous setting and outer measure spaces. Adv. Math. 285, 1155–1188 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Treil, S., Volberg, A.: Entropy conditions in two weight inequalities for singular integral operators. Adv. Math. 301, 499–548 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wang, G.: Sharp square-function inequalities for conditionally symmetric martingales. Trans. Am. Math. Soc. 328(1), 393–419 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gundy, R.F., Wheeden, R.L.: Weighted Integral Inequalities for the Nontangential Maximal function, Lusin area integral and Walsh Paley series. Stud. Math. 49, (1973/74), 107–124MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wilson, J.M.: Weighted norm inequalities for the continuous square function. Trans. Am. Math. Soc. 314(2), 661–692 (1989)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wilson, J.M.: \(L^{p}\) weighted norm inequalities for the square function, \(0<p<2\). Illinois J. Math. 33(3), 361–366 (1989)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Wittwer, J.: A sharp estimate on the norm of the martingale transform. Math. Res. Lett. 7(1), 1–12 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouseFrance
  2. 2.Department of Mathematical SciencesKent State UniversityKentUSA
  3. 3.Department of MathematicsBrown UniversityProvidenceUSA
  4. 4.Department of MathematicsMichigan State UniversityEast LansingUSA

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