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

Abstract

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.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 4(2), 765–778 (1975)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Álvarez, J., Milman, M.: Vector valued inequalities for strongly Calderón–Zygmund operators. Rev. Mat. Iberoam. 2, 405–426 (1986)

    MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Andersen, K., John, R.: Weighted inequalities for vector-valued maximal functions and singular integrals. Studia Math. 69(1), 19–31 (1981)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Benedek, A., Panzone, R.: The space \(L^p\) with mixed norm. Duke Math. J. 28, 301–324 (1961)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Caffarelli, L.: Elliptic second order equations. Rend. Sem. Mat. Fis. Milano. 58, 253–284 (1988)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Chuong, N.M.: Pseudodifferential operators and wavelets over real and p-adic fields. Springer, Berlin (2018)

    Google 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)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Cho, C.H., Yang, C.W.: Estimates for oscillatory strongly singular integral operators. J. Math. Anal. Appl. 362, 523–533 (2010)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93(1), 107–115 (1971)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Fefferman, C.: Inequality for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Federbush, P.: Navier and Stokes meet the wavelet. Commun. Math. Phys. 155, 219–248 (1993)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Grafakos, L.: Modern Fourier Analysis, 2nd edn. Springer, Berlin (2008)

    Google Scholar 

  20. 20.

    Ho, K.P.: Vector-valued maximal inequalities on weighted Orlicz–Morrey spaces. Tokyo J. Math. 36(2), 499–512 (2013)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Hunt, R.A.: On L(p, q) spaces. Enseign. Math. 12, 249–276 (1966)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Hirschman, I.I.: On multiplier transformations. Duke Math. J. 26, 221–242 (1959)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Indratno, S., Maldonado, D., Silwal, S.: A visual formalism for weights satisfying reverse inequalities. Expo. Math. 33, 1–29 (2015)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Kaneko, M., Yano, S.: Weighted norm inequalities for singular integrals. J. Math. Soc. Jpn. 27, 570–588 (1975)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282, 219–231 (2009)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Kokilashvili, V., Meskhi, A., Rafeiro, H.: Sublinear operators in generalized weighted Morrey spaces. Dokl. Math. 94, 558–560 (2016)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Lorentz, G.G.: Some new functional spaces. Ann. Math. 51(1), 37–55 (1950)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Lin, Y.: Strongly singular Calderón–Zygmund operator and commutator on Morrey type spaces. Acta Math. Sin. 23, 2097–2110 (2007)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Morrey, C.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)

    MathSciNet  MATH  Google 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)

    MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Pérez, C.: Endpoints for commutators of singular integral operators. J. Funct. Anal. 128, 163–185 (1995)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Pan, Y.: Hardy spaces and oscillatory singular integrals. Rev. Mat. Iberoam. 7, 55–64 (1991)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Ruiz, A., Vega, L.: Unique continuation for Schrödinger operators with potential in Morrey spaces. Publ. Mat. 35(1), 291–298 (1991)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Sawano, Y., Tanaka, H.: The Fatou property of block spaces. J. Math. Sci. Univ. Tokyo 22(3), 663–683 (2015)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Samko, N.: Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350(1), 56–72 (2009)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Shen, Z.: The periodic Schrödinger operators with potentials in the Morrey class. J. Funct. Anal. 193(2), 314–345 (2002)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Soria, F., Weiss, G.: A remark on singular integrals and power weights. Indiana Univ. Math. J. 43, 187–204 (1994)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    Google Scholar 

  46. 46.

    Sjölin, P.: \(L^p\) estimates for strongly singular convolution operators in \({\mathbb{R}}^n\). Ark. Mat. 14, 59–64 (1976)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Tanaka, H.: Two-weight norm inequalities on Morrey spaces. Ann. Acad. Sci. Fenn. Math. 40(2), 773–791 (2015)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Torchinsky, A.: Real Variable Methods in Harmonic Analysis. Academic Press, San Diego (1986)

    Google Scholar 

  49. 49.

    Tang, L.: Weighted norm inequalities for pseudodifferential operators with smooth symbols and their commutators. J. Funct. Anal. 262, 1603–1629 (2012)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Wang, D., Zhou, J., Chen, W.: Another characterizations of Muckenhoupt \(A_p\) class. Acta Math. Sci. 37, 1761–1774 (2017)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Wainger, S.: Special trigonometric series in k-dimensions. Mem. Am. Math. Soc. 56 (1965)

  53. 53.

    Zorko, C.T.: Morrey space. Proc. Am. Math. Soc. 98, 586–592 (1986)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nguyen Minh Chuong.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Minh Chuong, N., Van Duong, D. & Huu Dung, K. Maximal operators and singular integrals on the weighted Lorentz and Morrey spaces. J. Pseudo-Differ. Oper. Appl. 11, 201–228 (2020). https://doi.org/10.1007/s11868-019-00277-3

Download citation

Keywords

  • 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