The Multilinear Littlewood–Paley Operators with Minimal Regularity Conditions

Article
  • 55 Downloads

Abstract

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.

Keywords

Multilinear Sparse operators Littlewood–Paley functions Quantitative weighted estimates 

Mathematics Subject Classification

Primary 42B25 Secondary 42B20 

Notes

Acknowledgements

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

References

  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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bui, T.A., Hormozi, M.: Weighted bounds for multilinear square functions. Potential Anal. 46, 135–148 (2017)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, S., Wu, H., Xue, Q.: A note on multilinear Muckenhoupt classes for multiple weights. Stud. Math. 223, 1–18 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, X., Xue, Q., Yabuta, K.: On multilinear Littlewood-Paley operators. Nonlinear Anal. 115, 25–40 (2015)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Grafakos, L.: Classical and Modern Fourier Analysis. Prentice Hall, New Jersey (2004)MATHGoogle Scholar
  16. 16.
    Grafakos, L., Torres, R.H.: Multilinear Calderón-Zygmund theory. Adv. Math. 165, 124–164 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Grafakos, L., Liu, L., Pérez, C., Torres, R.H.: The multilinear strong maximal function. J. Geom. Anal. 21, 118–149 (2011)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Holmes, I., Lacey, M.T., Wick, B.D.: Commutators in the two-weight setting. Math. Ann. 367, 51–80 (2017)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hytönen, T.: The sharp weighted bound for general Calderón-Zygmund operators. Ann. Math. 175(3), 1473–1506 (2012)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE. 6, 777–818 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hytönen, T., Roncal, L., Tapiola, O.: Quantitative weighted estimates for rough homogeneous singular integrals. Israel J. Math. 218, 133–164 (2017)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lacey, M.: An elementary proof of the \(A_2\) bound. Israel J. Math. 217, 181–195 (2017)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lacey, M., Spencer, S.: Sparse bounds for oscillatory and random singular integrals. N. Y. J. Math. 23, 119–131 (2017)MathSciNetMATHGoogle Scholar
  30. 30.
    Lerner, A.: Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals. Adv. Math. 226, 3912–3926 (2011)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lerner, A.: A simple proof of the \(A_2\) conjecture. Int. Math. Res. Not. 14, 3159–3170 (2013)CrossRefMATHGoogle Scholar
  32. 32.
    Lerner, A.: On sharp aperture-weighted estimates for square functions. J. Four. Anal. Appl. 20, 784–800 (2014)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Lerner, A.: On pointwise estimates involving sparse operators. N. Y. J. Math. 22, 341–349 (2016)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    O’Neil, R.: Fractional integration in Orlicz spaces. Trans. Am. Math. Soc. 115, 300–328 (1965)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Pérez, C., Rivera-Ríos, I.P.: Borderline weighted estimates for commutators of singular integrals. Israel J. Math. 217, 435–475 (2017)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Yabuta, K.: On the kernel of pseudo-differential operators. Bull. Fac. Sci. Ibaraki Univ. A 18, 13–18 (1986)CrossRefMATHGoogle Scholar
  51. 51.
    Zorin-Kranich, P.: Intrinsic square functions with arbitrary aperture. arXiv:1605.02936

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

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

Personalised recommendations