Mixed estimates for singular integrals on weighted Hardy spaces


In this paper we give quantitative bounds for the norms of different kinds of singular integral operators on weighted Hardy spaces \(H_w^p\), where \(0<p\le 1\) and w is a weight in the Muckenhoupt \(A_{\infty }\) class. We deal with Fourier multiplier operators, Calderón–Zygmund operators of homogeneous type which are particular cases of the first ones, and, more generally, we study singular integrals of convolution type. In order to prove mixed estimates in the setting of weighted Hardy spaces, we need to introduce several characterizations of weighted Hardy spaces by means of square functions, Littlewood–Paley functions and the grand maximal function. We also establish explicit quantitative bounds depending on the weight w when switching between the \(H^p_w\)-norms defined by the Littlewood–Paley–Stein square function and its discrete version, and also by applying the mixed bound \(A_q-A_\infty \) result for the vector-valued extension of the Hardy–Littlewood maximal operator given in Buckley (Trans Am Math Soc 340(1):253–272, 1993).

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  1. 1.

    Buckley, S.M.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Am. Math. Soc. 340(1), 253–272 (1993)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Cejas, M.E., Li, K., Pérez, C., Rivera-Ríos,I.P.: A sparse approach to sharp weighted estimates for vector-valued operators. Sci. China Math. (in press). Preprint available at arXiv:1712.05781

  3. 3.

    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Extrapolation from \(A_\infty \) weights and applications. J. Funct. Anal. 213(2), 412–439 (2004)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Sharp weighted estimates for classical operators. Adv. Math. 229(1), 408–441 (2012)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Dragičević, O., Grafakos, L., Pereyra, M.C., Petermichl, S.: Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces. Publ. Mat. 49(1), 73–91 (2005)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Ding, Y., Yongsheng, H., Lu, G., Wu, X.: Boundedness of singular integrals on multiparameter weighted Hardy spaces \(H^p_w (\mathbb{R}^n\times \mathbb{R}^m)\). Potential Anal. 37(1), 31–56 (2012)

    MathSciNet  Google Scholar 

  7. 7.

    Dindoš, M., Wall, T.: The sharp \(A_p\) constant for weights in a reverse-Hölder class. Rev. Mat. Iberoam. 25(2), 559–594 (2009)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34(4), 777–799 (1985)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    García-Cuerva, J.: Weighted Hardy spaces. In: Harmonic Analysis in Euclidean Spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1

  10. 10.

    Gundy, R.F., Wheeden, R.L.: Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh–Paley series. Stud. Math. 49, 107–124 (1973/74)

  11. 11.

    Hytönen, T.P., Lacey, M.T., Pérez, C.: Sharp weighted bounds for the \(q\)-variation of singular integrals. Bull. Lond. Math. Soc. 45(3), 529–540 (2013)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Hart, J., Oliveira, L.: Hardy space estimates for limited ranges of Muckenhoupt weights. Adv. Math. 313, 803–838 (2017)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6(4), 777–818 (2013)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Hagelstein, P., Parissis, I.: Weighted Solyanik estimates for the Hardy-Littlewood maximal operator and embedding of \(\cal{A}_\infty \) into \(\cal{A}_p\). J. Geom. Anal. 26(2), 924–946 (2016)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Hytönen, T.P.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. Math. (2), 175(3):1473–1506 (2012)

  16. 16.

    Korey, M.B.: Ideal weights: asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation. J. Fourier Anal. Appl. 4(4–5), 491–519 (1998)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Kurtz, D.S., Wheeden, R.L.: Results on weighted norm inequalities for multipliers. Trans. Am. Math. Soc. 255, 343–362 (1979)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Lee, M.-Y.: Calderón-Zygmund operators on weighted Hardy spaces. Potential Anal. 38(3), 699–709 (2013)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Lerner, A.K.: On some weighted norm inequalities for Littlewood-Paley operators. Illinois J. Math. 52(2), 653–666 (2008)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Lee, M.-Y., Lin, C.-C.: The molecular characterization of weighted Hardy spaces. J. Funct. Anal. 188(2), 442–460 (2002)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Lerner, A.K., Ombrosi, S., Pérez, C.: Sharp \(A_1\) bounds for Calderón–Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden. Int. Math. Res. Not. IMRN, (6):Art. ID rnm161, 11 (2008)

  22. 22.

    Lu, G., Zhu, Y.: Bounds of singular integrals on weighted Hardy spaces and discrete Littlewood–Paley analysis. J. Geom. Anal. 22(3), 666–684 (2012)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Mitsis, T.: Embedding \(B_\infty \) into Muckenhoupt classes. Proc. Am. Math. Soc. 133(4), 1057–1061 (2005)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Meda, S., Sjögren, P., Vallarino, M.: Atomic decompositions and operators on Hardy spaces. Rev. Un. Mat. Argentina 50(2), 15–22 (2009)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Petermichl, S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical \(A_p\) characteristic. Am. J. Math. 129(5), 1355–1375 (2007)

    MATH  Google Scholar 

  26. 26.

    Petermichl, S.: The sharp weighted bound for the Riesz transforms. Proc. Am. Math. Soc. 136(4), 1237–1249 (2008)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Politis, A.: Sharp results on the relation between weight spaces and BMO. ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.), The University of Chicago

  28. 28.

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

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Petermichl, S., Volberg, A.: Heating of the Ahlfors–Beurling operator: weakly quasiregular maps on the plane are quasiregular. Duke Math. J. 112(2), 281–305 (2002)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Strömberg, J.-O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Mathematics, vol. 1381. Springer, Berlin (1989)

    Google Scholar 

  31. 31.

    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J. (1970)

  32. 32.

    Wik, I.: On Muckenhoupt’s classes of weight functions. Stud. Math. 94(3), 245–255 (1989)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Wilson, J.M.: Weighted inequalities for the square function. In: Commutative harmonic analysis (Canton, NY, 1987), volume 91 of Contemp. Math., pp. 299–305. Amer. Math. Soc., Providence (1989)

  34. 34.

    Wilson, J.M.: Chanillo-Wheeden inequalities for \(0<p\le 1\). J. Lond. Math. Soc. (2), 41(2):283–294 (1990)

  35. 35.

    Wilson, M.: Weighted Littlewood–Paley Theory and Exponential-Square Integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2008)

    Google Scholar 

  36. 36.

    Wang, H., Liu, H.: The intrinsic square function characterizations of weighted Hardy spaces. Illinois J. Math. 56(2), 367–381 (2012)

    MathSciNet  MATH  Google Scholar 

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María Eugenia wants to mention that the study of these estimates was pointed out by Sheldy Ombrosi, her supervisor during postdoctoral research, which was done under a fellowship granted by CONICET, Argentina. Also, she would like to thank Carlos Pérez, Cristina Pereyra and Michael Wilson for their help and interest when contacting them.


First author was supported by Universidad de Buenos Aires (Grant No. 20020120100050), by Agencia Nacional de Promoción Científica y Tecnológica (Grant No. PICT 2014-1771) and by Universidad Nacional de La Plata (Grant No. UNLP 11/X752 and UNLP 11/X805). Second author was supported by Universidad Nacional del Litoral (Grants No. CAI+D 2015-026 and CAI+D 2015-066).

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Correspondence to Estefanía Dalmasso.

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Cejas, M.E., Dalmasso, E. Mixed estimates for singular integrals on weighted Hardy spaces. Rev Mat Complut 33, 745–766 (2020). https://doi.org/10.1007/s13163-019-00331-0

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  • Weighted Hardy spaces
  • Singular integrals
  • Mixed estimates
  • Calderón–Zygmund operators
  • Fourier multipliers

Mathematics Subject Classification

  • 42B30
  • 42B20
  • 42B15
  • 42B25