Maximal operators and singular integrals on the weighted Lorentz and Morrey spaces

  • Nguyen Minh ChuongEmail author
  • Dao Van Duong
  • Kieu Huu Dung


In this paper, we first give some new characterizations of Muckenhoupt type weights through establishing the boundedness of maximal operators on the weighted Lorentz and Morrey spaces. Secondly, we establish the boundedness of sublinear operators including many interesting in harmonic analysis and its commutators on the weighted Morrey spaces. Finally, as an application, the boundedness of strongly singular integral operators and commutators with symbols in BMO space are also given.


Maximal function Sublinear operator Strongly singular integral Commutator \(A_p\) weight \(A(p{, } 1)\) weight \(A_p(\varphi )\) weight BMO space Lorentz spaces Morrey spaces 

Mathematics Subject Classification

42B20 42B25 42B99 



The authors are grateful to the anonymous referee for the valuable suggestions and comments which lead to the improvement of the paper. The authors are supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 101.02-2014.51.


  1. 1.
    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 4(2), 765–778 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Álvarez, J., Milman, M.: Vector valued inequalities for strongly Calderón–Zygmund operators. Rev. Mat. Iberoam. 2, 405–426 (1986)CrossRefzbMATHGoogle Scholar
  3. 3.
    Alvarez, J., Guzmán-Partida, M., Lakey, J.: Spaces of bounded \(\lambda \)-central mean oscillation, Morrey spaces, and \(\lambda \)-central Carleson measures. Collect. Math. 51, 1–47 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Andersen, K., John, R.: Weighted inequalities for vector-valued maximal functions and singular integrals. Studia Math. 69(1), 19–31 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benedek, A., Panzone, R.: The space \(L^p\) with mixed norm. Duke Math. J. 28, 301–324 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caffarelli, L.: Elliptic second order equations. Rend. Sem. Mat. Fis. Milano. 58, 253–284 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chung, H.M., Hunt, R.A., Kurtz, D.S.: The Hardy–Littlewood maximal function on \(L(p, q)\) spaces with weights. Indiana Univ. Math. J. 31(1), 109–120 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chuong, N.M.: Pseudodifferential operators and wavelets over real and p-adic fields. Springer, Berlin (2018)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chuong, N.M., Duong, D.V., Dung, K.H.: Vector valued maximal Carleson type operators on the weighted Lorentz spaces (2017), arXiv:1707.00092v1
  10. 10.
    Coifman, R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51(3), 241–250 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cho, C.H., Yang, C.W.: Estimates for oscillatory strongly singular integral operators. J. Math. Anal. Appl. 362, 523–533 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Duoandiotxea, J., Rosenthal, M.: Extension and boundedness of operators on Morrey spaces from extrapolation techniques and embeddings. J. Geom. Anal. 28(4), 3081–3108 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fan, D., Lu, S., Yang, D.: Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients. Georgian Math. J. 5(5), 425–440 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93(1), 107–115 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fefferman, C.: Inequality for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Federbush, P.: Navier and Stokes meet the wavelet. Commun. Math. Phys. 155, 219–248 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guliyev, V.S., Aliyev, S.S., Karaman, T., Shukurov, P.S.: Boundedness of sublinear operators and commutators on generalized Morrey spaces. Integr. Equ. Oper. Theory 71, 327–355 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gurbuz, F.: Weighted Morrey and weighted fractional Sobolev–Morrey spaces estimates for a large class of pseudo-differential operators with smooth symbols. J. Pseudo-Differ. Oper. Appl. 7, 595–607 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grafakos, L.: Modern Fourier Analysis, 2nd edn. Springer, Berlin (2008)zbMATHGoogle Scholar
  20. 20.
    Ho, K.P.: Vector-valued maximal inequalities on weighted Orlicz–Morrey spaces. Tokyo J. Math. 36(2), 499–512 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hunt, R.A.: On L(p, q) spaces. Enseign. Math. 12, 249–276 (1966)zbMATHGoogle Scholar
  22. 22.
    Hytönen, T., Pérez, C., Rela, E.: Sharp reverse Hölder property for \(A_\infty \) weights on spaces of homogeneous type. J. Funct. Anal. 263, 3883–3899 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hirschman, I.I.: On multiplier transformations. Duke Math. J. 26, 221–242 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Indratno, S., Maldonado, D., Silwal, S.: A visual formalism for weights satisfying reverse inequalities. Expo. Math. 33, 1–29 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kaneko, M., Yano, S.: Weighted norm inequalities for singular integrals. J. Math. Soc. Jpn. 27, 570–588 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282, 219–231 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kokilashvili, V., Meskhi, A., Rafeiro, H.: Sublinear operators in generalized weighted Morrey spaces. Dokl. Math. 94, 558–560 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lorentz, G.G.: Some new functional spaces. Ann. Math. 51(1), 37–55 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Li, J., Lu, S.: \(L^p\) estimates for multilinear operators of strongly singular integral operators. Nagoya Math. J. 181, 41–62 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lin, Y.: Strongly singular Calderón–Zygmund operator and commutator on Morrey type spaces. Acta Math. Sin. 23, 2097–2110 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mazzucato, A.L.: Besov–Morrey spaces: function space theory and applications to non-linear PDE. Trans. Am. Math. Soc. 355(4), 1297–1364 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mastylo, M., Sawano, Y., Tanaka, H.: Morrey type space and its Köthe dual space. Bull. Malays. Math. Soc. 41, 1181–1198 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Morrey, C.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Nakai, E., Tomita, N., Yabuta, K.: Density of the set of all infinitely differentiable functions with compact support in weighted Sobolev spaces. Sci. Math. Jpn. Online. 10, 39–45 (2004)zbMATHGoogle Scholar
  36. 36.
    Nakai, E.: Hardy–Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Nakamura, S., Sawano, Y.: The singular integral operator and its commutator on weighted Morrey spaces. Collect. Math. 68(2), 145–174 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Pérez, C.: Endpoints for commutators of singular integral operators. J. Funct. Anal. 128, 163–185 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pan, Y.: Hardy spaces and oscillatory singular integrals. Rev. Mat. Iberoam. 7, 55–64 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ruiz, A., Vega, L.: Unique continuation for Schrödinger operators with potential in Morrey spaces. Publ. Mat. 35(1), 291–298 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Sawano, Y., Tanaka, H.: The Fatou property of block spaces. J. Math. Sci. Univ. Tokyo 22(3), 663–683 (2015)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Samko, N.: Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350(1), 56–72 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Shen, Z.: The periodic Schrödinger operators with potentials in the Morrey class. J. Funct. Anal. 193(2), 314–345 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Soria, F., Weiss, G.: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43, 187–204 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  46. 46.
    Sjölin, P.: \(L^p\) estimates for strongly singular convolution operators in \({\mathbb{R}}^n\). Ark. Mat. 14, 59–64 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Tanaka, H.: Two-weight norm inequalities on Morrey spaces. Ann. Acad. Sci. Fenn. Math. 40(2), 773–791 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Torchinsky, A.: Real Variable Methods in Harmonic Analysis. Academic Press, San Diego (1986)zbMATHGoogle Scholar
  49. 49.
    Tang, L.: Weighted norm inequalities for pseudodifferential operators with smooth symbols and their commutators. J. Funct. Anal. 262, 1603–1629 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Taylor, M.E.: Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations. Commun. Partial Differ. Equ. 17, 1407–1456 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Wang, D., Zhou, J., Chen, W.: Another characterizations of Muckenhoupt \(A_p\) class. Acta Math. Sci. 37, 1761–1774 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Wainger, S.: Special trigonometric series in k-dimensions. Mem. Am. Math. Soc. 56 (1965)Google Scholar
  53. 53.
    Zorko, C.T.: Morrey space. Proc. Am. Math. Soc. 98, 586–592 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Mathematics, Vietnamese Academy of Science and TechnologyHanoiVietnam
  2. 2.School of Mathematics, Mientrung University of Civil EngineeringPhu YenVietnam
  3. 3.School of Mathematics, University of Transport and CommunicationsHanoiVietnam

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