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

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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.

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References

  1. Bennett, C., Sharpley, R.: Interpolation of Operators, Pure and Applied Mathematics, vol. 129. Academic Press Inc., Boston, MA (1988)

    MATH  Google Scholar 

  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. 294–306

  3. Burkholder, D.L.: Martingale transforms. Ann. Math. Stat. 37, 1494–1504 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  6. Hytonen, T., Perez, C.: Sharp weighted bounds involving \(A_{\infty }\). Anal. PDE 6(4), 777–818 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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), 107–124

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lacey, M.T.: An elementary proof of the \(A_2\) bound. Israel J. Math. 217(1), 181–195 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lacey, M.T., Li, K.: On \(A_p\)-\(A_\infty \) type estimates for square functions. Math. Z. 284(3–4), 1211–1222 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  12. Petermichl, S., Pott, S.: An estimate for weighted Hilbert transform via square functions. Trans. Am. Math. Soc. 354(4), 1699–1703 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Treil, S., Volberg, A.: Entropy conditions in two weight inequalities for singular integral operators. Adv. Math. 301, 499–548 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, G.: Sharp square-function inequalities for conditionally symmetric martingales. Trans. Am. Math. Soc. 328(1), 393–419 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  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), 107–124

  17. Wilson, J.M.: Weighted norm inequalities for the continuous square function. Trans. Am. Math. Soc. 314(2), 661–692 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wilson, J.M.: \(L^{p}\) weighted norm inequalities for the square function, \(0<p<2\). Illinois J. Math. 33(3), 361–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wittwer, J.: A sharp estimate on the norm of the martingale transform. Math. Res. Lett. 7(1), 1–12 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062, Toulouse, France

    K. Domelevo & S. Petermichl

  2. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    P. Ivanisvili

  3. Department of Mathematics, Brown University, Providence, RI, 02912, USA

    S. Treil

  4. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    A. Volberg

Authors
  1. K. Domelevo
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  2. P. Ivanisvili
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  3. S. Petermichl
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  4. S. Treil
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  5. A. Volberg
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Corresponding author

Correspondence to S. Petermichl.

Additional information

Communicated by Loukas Grafakos.

Komla Domelevo and Stefanie Petermichl were supported by ERC project CHRiSHarMa no. DLV-682402. Sergei Treil was supported by the NSF Grants DMS-1600139. Alexander Volberg was supported by NSF Grant DMS-1600065. The paper was written while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring semester of 2017, supported by the National Science Foundation under Grant no. 1440140.

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Domelevo, K., Ivanisvili, P., Petermichl, S. et al. On the failure of lower square function estimates in the non-homogeneous weighted setting. Math. Ann. 374, 1923–1952 (2019). https://doi.org/10.1007/s00208-018-1787-4

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  • DOI: https://doi.org/10.1007/s00208-018-1787-4

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