The Multilinear Littlewood–Paley Operators with Minimal Regularity Conditions

  • Mingming CaoEmail author
  • Kôzô Yabuta


We investigate the quantitative weighted estimates for a large class of the multilinear Littlewood–Paley square operators. Our kernels satisfy the minimal regularity assumption, called \(L^r\)-Hörmander condition. We respectively establish the pointwise sparse domination for the multilinear square functions and their iterated commutators. Based on them, we obtain the strong type quantitative bounds and endpoint estimates. We recover lots of known weighted inequalities for Littlewood–Paley operators. Significantly, the approach is dyadic, quite elementary and simpler than that presented previously.


Multilinear Sparse operators Littlewood–Paley functions Quantitative weighted estimates 

Mathematics Subject Classification

Primary 42B25 Secondary 42B20 



The authors would like to thank the referee for valuable suggestions which have improved the quality of this paper.


  1. 1.
    Benea, C., Bernicot, F., Luque, T.: Sparse bilinear forms for Bochner-Riesz multipliers and applications. Trans. Lond. Math. Soc. 4(1), 110–128 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bernicot, F., Frey, D., Petermichl, S.: Sharp weighted norm estimates beyond Calderón-Zygmund theory. Anal. PDE 9(5), 1079–1113 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Buckley, S.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Am. Math. Soc. 340(1), 253–272 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bui, T.A., Duong, X.T.: Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers. Bull. Sci. Math. 137(1), 63–75 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bui, T.A., Hormozi, M.: Weighted bounds for multilinear square functions. Potential Anal. 46, 135–148 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, S., Wu, H., Xue, Q.: A note on multilinear Muckenhoupt classes for multiple weights. Stud. Math. 223, 1–18 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, X., Xue, Q., Yabuta, K.: On multilinear Littlewood-Paley operators. Nonlinear Anal. 115, 25–40 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chung, D., Pereyra, M.C., Pérez, C.: Sharp bounds for general commutators on weighted Lebesgue spaces. Trans. Am. Math. Soc. 364, 1163–1177 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cladek, L., Ou, Y.: Sparse domination of Hilbert transforms along curves. arXiv:1704.07810
  10. 10.
    Conde-Alonso, J., Rey, G.: A pointwise estimate for positive dyadic shifts and some applications. Math. Ann. 365, 1111–1135 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Conde-Alonso, J., Culiuc, A., Di Plinio, F., Ou, Y.: A sparse domination principle for rough singular integrals. Anal. PDE 10, 1255–1284 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Culiuc, A., Di Plinio, F., Ou, Y.: Domination of multilinear singular integrals by positive sparse forms. arXiv:1603.05317
  13. 13.
    Damián, W., Hormozi, M., Li, K.: New bounds for bilinear Calderón-Zygmund operators and applications. Rev. Mat. Iberoam (to appear)Google Scholar
  14. 14.
    Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129, 137–193 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grafakos, L.: Classical and Modern Fourier Analysis. Prentice Hall, New Jersey (2004)zbMATHGoogle Scholar
  16. 16.
    Grafakos, L., Torres, R.H.: Multilinear Calderón-Zygmund theory. Adv. Math. 165, 124–164 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grafakos, L., Liu, L., Pérez, C., Torres, R.H.: The multilinear strong maximal function. J. Geom. Anal. 21, 118–149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hänninen, T.S.: Remark on dyadic pointwise domination and median oscillation decomposition. arXiv:1502.05942v1
  19. 19.
    He, S., Xue, Q., Mei, T., Yabuta, K.: Existence and boundedness of multilinear Littlewood-Paley operators on Campanato spaces. J. Math. Anal. Appl. 432, 86–102 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Holmes, I., Lacey, M.T., Wick, B.D.: Commutators in the two-weight setting. Math. Ann. 367, 51–80 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hormozi, M., Li, K.: \(A_p\)-\(A_{\infty }\) estimates for general multilinear sparse operators. arXiv:1508.06105v3
  22. 22.
    Hu, G., Li, D.: A Cotlar type inequality for the multilinear singular integral operators and its applications. J. Math. Anal. Appl. 290, 639–653 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hytönen, T.: The sharp weighted bound for general Calderón-Zygmund operators. Ann. Math. 175(3), 1473–1506 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hytönen, T.: The \(A_2\) Theorem: Remarks and Complements. Contemporary Mathematics, vol. 612, pp. 91–106. American Mathematical Society, Providence (2014)Google Scholar
  25. 25.
    Hytönen, T.P., Lacey, M.T., Pérez, C.: Sharp weighted bounds for the \(q\)-variation of singular integrals. Bull. Lond. Math. Soc. 45, 529–540 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE. 6, 777–818 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hytönen, T., Roncal, L., Tapiola, O.: Quantitative weighted estimates for rough homogeneous singular integrals. Israel J. Math. 218, 133–164 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lacey, M.: An elementary proof of the \(A_2\) bound. Israel J. Math. 217, 181–195 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lacey, M., Spencer, S.: Sparse bounds for oscillatory and random singular integrals. N. Y. J. Math. 23, 119–131 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lerner, A.: Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals. Adv. Math. 226, 3912–3926 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lerner, A.: A simple proof of the \(A_2\) conjecture. Int. Math. Res. Not. 14, 3159–3170 (2013)CrossRefzbMATHGoogle Scholar
  32. 32.
    Lerner, A.: On sharp aperture-weighted estimates for square functions. J. Four. Anal. Appl. 20, 784–800 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lerner, A.: On pointwise estimates involving sparse operators. N. Y. J. Math. 22, 341–349 (2016)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Lerner, A., Nazarov, F.: Intuitive dyadic calculus: the basics. arXiv:1508.05639
  35. 35.
    Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 220, 1222–1264 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lerner, A.K., Ombrosi, S., Rivera-Ríos, I.P.: On pointwise and weighted estimates for commutators of Calderón-Zygmund operators. Adv. Math. 319, 153–181 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Li, K.: Sparse domination theorem for multilinear singular integral operators with \(L^r\)-Hörmander condition. Michigan Math. J. (to appear)Google Scholar
  38. 38.
    Li, K., Sun, W.: Weak and strong type weighted estimates for multilinear Calderón-Zygmund operators. Adv. Math. 254, 736–771 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Li, K., Moen, K., Sun, W.: The sharp weighted bound for multilinear maximal functions and Calderón-Zygmund operators. J. Four. Anal. Appl. 20, 751–765 (2014)CrossRefzbMATHGoogle Scholar
  40. 40.
    Lorente, M., Riveros, M.S., de la Torre, A.: Weighted estimates for singular integral operators satisfying Hörmander’s conditions of Young type. J. Fourier Anal. Appl. 11, 497–509 (2015)CrossRefzbMATHGoogle Scholar
  41. 41.
    Martell, J., Pérez, C., Trujillo-González, R.: Lack of natural weighted estimates for some singular integral operators. Trans. Am. Math. Soc. 357, 385–396 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    O’Neil, R.: Fractional integration in Orlicz spaces. Trans. Am. Math. Soc. 115, 300–328 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Pérez, C., Rivera-Ríos, I.P.: Borderline weighted estimates for commutators of singular integrals. Israel J. Math. 217, 435–475 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Pérez, C., Pradolini, G., Torres, R.H., Trujillo-González, R.: End-point estimates for iterated commutators of multilinear singular integrals. Bull. Lond. Math. Soc. 46, 26–42 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Shi, S., Xue, Q., Yabuta, K.: On the boundedness of multilinear Littlewood-Paley \(g_{\lambda }^*\) function. J. Math. Pures Appl. 101, 394–413 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Si, Z., Xue, Q., Yabuta, K.: On the bilinear square Fourier multiplier operators and related multilinear square functions. Sci. China Math. 60, 1477–1502 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Stein, E.M.: Singular integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton, NJ (1970)Google Scholar
  48. 48.
    Xue, Q., Yan, J.: On multilinear square function and its applications to multilinear Littlewood-Paley operators with non-convolution type kernels. J. Math. Anal. Appl. 422, 1342–1362 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Xue, Q., Peng, X., Yabuta, K.: On the theory of multilinear Littlewood-Paley \(g\)-function. J. Math. Soc. Jpn. 67, 535–559 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Yabuta, K.: On the kernel of pseudo-differential operators. Bull. Fac. Sci. Ibaraki Univ. A 18, 13–18 (1986)CrossRefzbMATHGoogle Scholar
  51. 51.
    Zorin-Kranich, P.: Intrinsic square functions with arbitrary aperture. arXiv:1605.02936

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Authors and Affiliations

  1. 1.Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China
  2. 2.Research Center for Mathematical SciencesKwansei Gakuin UniversitySandaJapan

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