Quantitative Weighted Estimates for Rubio de Francia’s Littlewood–Paley Square Function

  • Rahul Garg
  • Luz RoncalEmail author
  • Saurabh Shrivastava


We consider the Rubio de Francia’s Littlewood–Paley square function associated with an arbitrary family of intervals in \({\mathbb {R}}\) with finite overlapping. Quantitative-weighted estimates are obtained for this operator. The linear dependence on the characteristic of the weight \([w]_{A_{p/2}}\) turns out to be sharp for \(3\le p<\infty \), whereas the sharpness in the range \(2<p<3\) remains as an open question. Weighted weak-type estimates in the endpoint \(p=2\) are also provided. The results arise as a consequence of a sparse domination shown for these operators, obtained by suitably adapting the ideas coming from Benea (Vector-valued extensions for singular bilinear operators and applications, PhD thesis, Cornell University, 2015) and Culiuc et al. (J Lond Math Soc (2) 98(2):369–392, 2018).


Rubio de Francia’s Littlewood–Paley square function Sparse domination Weighted norm inequalities 

Mathematics Subject Classification

Primary 42B20 Secondary 42B25 42B35 



This work was initiated during the visit of the second author to R. Garg and S. Shrivastava at IISER Bhopal in July 2018. The second author is very grateful for the kind hospitality. The authors would also like to thank Andrei Lerner for several discussions and suggestions related to this project and to Camil Muscalu for offering clarifications about the helicoidal method. We are also greatly indebted to the referee for helpful and valuable remarks. The first author was supported in part by the INSPIRE Faculty Award from the Department of Science and Technology (DST), Government of India. The second author was supported by the Basque Government through BERC 2018–2021 program, by Spanish Ministry of Science, Innovation and Universities through BCAM Severo Ochoa accreditation SEV-2017-2018 and the project MTM2017-82160-C2-1-P funded by AEI/FEDER, UE, and by 2017 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation. The Foundation accepts no responsibility for the opinions, statements and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors. The third author was supported by Science and Engineering Research Board (SERB), Government of India, under the Grant MATRICS: MTR/2017/000039/Math.


  1. 1.
    Bakas, O.: Endpoint mapping properties of the Littlewood–Paley square function. Colloq. Math. 157(1), 1–15 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Beltran, D., Cladek, L.: Sparse bounds for pseudodifferential operators. J. Anal. Math. ArXiv:1711.02339
  3. 3.
    Benea, C.M.: Vector-valued extensions for singular bilinear operators and applications. PhD thesis, Cornell University (2015)Google Scholar
  4. 4.
    Benea, C.M., Bernicot, F.: A bilinear Rubio de Francia inequality for arbitrary squares. Forum Math. Sigma 4 , e26 (2016)Google Scholar
  5. 5.
    Benea, C.M., Muscalu, C.: Sparse domination via the helicoidal method. arXiv:1707.05484
  6. 6.
    Benea, C.M., Muscalu, C.: The Helicoidal Method, Operator Theory: Themes and Variations. Theta Ser. Adv. Math., vol. 20, pp. 45–96. Theta, Bucharest (2018)Google Scholar
  7. 7.
    Bernicot, F., Frey, D., Petermichl, S.: Sharp weighted norm estimates beyond Calderón–Zygmund theory. Anal. PDE 9(5), 1079–1113 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bernicot, F.: \(L^p\) estimates for non smooth bilinear Littlewood–Paley square functions on \({\mathbb{R}}\). Math. Ann. 351(1), 1–49 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bernicot, F., Shrivastava, S.: Boundedness of smooth bilinear square functions and applications to some bilinear pseudo-differential operators. Indiana Univ. Math. J. 60(1), 233–268 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bourgain, J.: On square functions on the trigonometric system. Bull. Soc. Math. Belg. Sér. B 37(1), 20–26 (1985)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bourgain, J.: On the Behavior of the Constant in the Littlewood–Paley Inequality. Geometric Aspects of Functional Analysis (1987–88). Lecture Notes in Math., vol. 1376, pp. 202–208. Springer, Berlin (1989)Google Scholar
  12. 12.
    Carleson, L.: On the Littlewood–Paley theorem. Inst. Mittag-Leffler Report (1967)Google Scholar
  13. 13.
    Chen, W., Culiuc, A., Di Plinio, F., Lacey, M., Ou, Y.: Endpoint sparse bounds for Walsh–Fourier multipliers of Marcienkiewicz type. arXiv:1805.06060
  14. 14.
    Córdoba, A.: Some remarks on the Littlewood–Paley theory. Rend. Circ. Mat. Palermo. (2) 1, 75–80 (1981)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cowling, M., Terence, T.: Some light on Littlewood–Paley theory. Math. Ann. 321(4), 885–888 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Cruz-Uribe, D., Martell, J.M., Ṕerez, C.: Sharp weighted estimates for classical operators. Adv. Math. 229(1), 408–441 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Culiuc, A., Di Plinio, F., Ou, Y.: Domination of multilinear singular integrals by positive sparse forms. J. Lond. Math. Soc. (2) 98(2), 369–392 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Di Plinio, F., Ou, Y.: A modulation invariant Carleson embedding theorem outside local \(L^2\). J. Anal. Math. 135(2), 675–711 (2018)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Do, Y., Thiele, C.: \(L^p\) theory for outer measures and two themes of Lennart Carleson united. Bull. Am. Math. Soc. (N.S.) 52, 249–296 (2015)zbMATHGoogle Scholar
  20. 20.
    Duoandikoetxea, J.: Fourier Analysis. (English Summary). Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence, RI (2001)zbMATHGoogle Scholar
  21. 21.
    Duoandikoetxea, J.: Extrapolation of weights revisited: new proofs and sharp bounds. J. Funct. Anal. 260(6), 1886–1901 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Frey, D., Nieraeth, B.: Weak and strong type \(A_1-A_{\infty }\) estimates for sparsely dominated operators. J. Geom. Anal. 29(1), 247–282 (2019)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hytönen, T.P.: The sharp weighted bound for general Calderón–Zygmund operators. Ann. Math. 175(3), 1473–1506 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hytönen, T.P., Roncal, L., Tapiola, O.: Quantitative weighted estimates for rough homogeneous singular integrals. Israel J. Math. 218(1), 133–164 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Journé, J.-L.: Calderón–Zygmund operators on product spaces. Rev. Mat. Iberoam. 1(3), 55–91 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Król, S.: Fourier multipliers on weighted \(L^p\) spaces. Math. Res. Lett. 21(4), 807–830 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kurtz, D.S.: Littlewood–Paley and multiplier theorems on weighted \(L_p\) spaces. Trans. Am. Math. Soc. 259(1), 235–254 (1980)zbMATHGoogle Scholar
  28. 28.
    Lacey, M.T.: Issues related to Rubio de Francia’s Littlewood–Paley inequality. New York J. Math. NYJM Monographs, vol. 2. State University of New York, University at Albany, Albany, NY (2007)Google Scholar
  29. 29.
    Lacey, M.T., Thiele, C.: \(L^p\) estimates on the bilinear Hilbert transform for \(2<p<\infty \). Ann. Math. (2) 146(3), 693–724 (1997)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lacey, M.T., Thiele, C.: On Calderón’s conjecture. Ann. Math. (2) 149(2), 475–496 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Lerner, A.: Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals. Adv. Math. 226, 3912–3926 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Lerner, A.: Quantitative weighted estimates for the Littlewood–Paley square function and Marcinkiewicz multipliers. Math. Res. Lett. ArXiv:1803.06981
  33. 33.
    Lerner, A., Nazarov, F.: Intuitive dyadic calculus: the basics. Expo. Math. 37(3), 225–265 (2019)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Li, K.: Two weight inequalities for bilinear forms. Collect. Math. 68, 129–144 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Littlewood, J.E., Paley, R.E.A.C.: Theorems on Fourier series and power series, II. Proc. Lond. Math. Soc. 42, 52–89 (1926)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Luque, T., Pérez, C., Rela, E.: Optimal exponents in weighted estimates without examples. Math. Res. Lett. 22, 183–201 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Muscalu, C., Tao, T., Thiele, C.: \({L}^p\) estimates for the biest I: the Walsh case. Math. Ann. 329, 401–426 (2004)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Muscalu, C., Tao, T., Thiele, C.: \({L}^p\) estimates for the biest II: the Fourier case. Math. Ann. 329, 427–461 (2004)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Oberlin, R., Seeger, A., Tao, T., Thiele, C., Wright, J.: A variation norm Carleson theorem. J. Eur. Math. Soc. (JEMS) 14(2), 421–464 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Rubio de Francia, J.L.: Estimates for some square functions of Littlewood–Paley type. Publ. Sec. Mat. Univ. Autòn. Barc. 27(2), 81–108 (1983)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Rubio de Francia, J.L.: A Littlewood–Paley inequality for arbitrary intervals. Rev. Mat. Iberoam. 1(2), 1–14 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Sjölin, P.: A note on Littlewood–Paley decompositions with arbitrary intervals. J. Approx. Theory 48(3), 328–334 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Soria, F.: A note on a Littlewood–Paley inequality for arbitrary intervals in \({\mathbb{R}}^2\). J. Lond. Math. Soc. (2) 36(1), 137–142 (1987)zbMATHGoogle Scholar
  44. 44.
    Thiele, C.: Wave Packet Analysis. CBMS Regional Conference Series in Mathematics, vol. 105 (Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Science Education and Research BhopalBhopalIndia
  2. 2.BCAM - Basque Center for Applied MathematicsBilbaoSpain
  3. 3.Ikerbasque, Basque Foundation for ScienceBilbaoSpain

Personalised recommendations