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Function characterizations via commutators of Hardy operator

Abstract

This paper is a summary of the research on the characterizations of central function spaces by the author and his collaborators in the past ten years. More precisely, the author gives some characterizations of central Campanato spaces via the boundedness and compactness of commutators of Hardy operator.

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References

  1. 1.

    Adams D R, Xiao J. Morrey spaces in harmonic analysis. Ark Mat, 2012, 50: 201–230

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Adams D R, Xiao J. Regularity of Morrey commutators. Trans Amer Math Soc, 2012, 364: 4801–4818

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Alvarez J, Guzmán-Partida M, Lakey J. Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures. Collect Math, 2000, 51: 1–47

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Anderson K, Muckenhoupt B. Weighted weak type Hardy inequalities with application to Hilbert transforms and maximal functions. Studia Math, 1982, 72: 9–26

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Beatrous F, Li S Y, On the boundedness and compactness of operators of Hankel type. J Funct Anal, 1993, 111: 350–379

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Branmanti M, Cerutti M. \(W_p^{1,2}\)-solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients. Comm Partial Differential Equations. 1993, 18: 1735–1763

    MathSciNet  Article  Google Scholar 

  7. 7.

    Campanato S. Proprietàdi Hölderianità di alcune classi di funzioni. Ann Sc Norm Super Pisa, 1963, 17: 173–188

    MATH  Google Scholar 

  8. 8.

    Chanillo S. A note on commutators. Indiana Univ Math J, 1982, 31: 7–16

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Chen Y P, Ding Y. Compactness of the commutators of parabolic singular integrals. Sci China Math, 2010, 53: 2633–2648

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Chen Y P, Ding Y. Compactness of commutators of Riesz potential on Morrey spaces. Potential Anal, 2009, 30: 301–313

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Chen Y P, Ding Y. Compactness of commutators for singular integrals on Morrey spaces. Canad J Math, 2012, 64: 257–281

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Chiarenza F, Frasca M, Longo P. W2,p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans Amer Math Soc, 1993, 336: 841–853

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Christ M, Grafakos L. Best constants for two nonconvolution inequalities. Proc Amer Math Soc, 1995, 123: 1687–1693

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Coifman R R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann of Math, 1976, 103: 611–635

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Cruz D, Neugebauer C J. The structure of the reverse Hölder classes. Trans Amer Math Soc, 1995, 345: 2941–2960

    MATH  Google Scholar 

  16. 16.

    Deng D G, Duong X T, Yan L X. A characterization of Morrey-Campanato spaces. Math Z, 2005, 250: 641–655

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Ding Y. A characterization of BMO via commutators for some operators. Northeast Math, 1997, 13: 422–432

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Ding Y, Mei T. Boundedness and compactness for the commutators of bilinear operators on Morrey spaces. Potential Anal, 2015, 42: 717–748

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Duong X T, Xiao J, Yan L X. Old and new Morrey spaces with heat kernel bounds. J Fourier Anal Appl, 2007, 13: 87–111

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Fan D S, Lu S Z, Yang D C. Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients. Georgian Math J, 1998, 5: 425–440

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Faris W. Weak Lebesgue spaces and quantum mechanical binding. Duke Math J, 1976, 43: 365–373

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Fazio G D, Ragusa M A. Interior estimates in Morrey spaces for strongly solutions to nondivergence form equations with discontinuous coefficients. J Funct Anal, 1993, 112: 241–256

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Fefferman C. The uncertainty principle. Bull Amer Math Soc, 1983, 9: 129–206

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Fu Z W, Liu Z G, Lu S Z, Wang H B. Characterization for commutators of n-dimensional fractional Hardy operators. Sci China Ser A, 2007, 50: 1418–1426

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Fu Z W, Lu S Z. Commutators of generalized Hardy operators. Math Nachr, 2009, 282: 832–845

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Fu Z W, Wu Q Y, Lu S Z. Sharp estimates of p-adic Hardy and Hardy-Littlewood-Pólya operators. Acta Math Sin (Engl Ser), 2013, 29: 137–150

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    García-Cuerva J. Hardy spaces and Beurling algebras. J Lond Math Soc, 1989, 39: 499–513

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Gilbarg D, Trudinger N. Elliptic Partial Differential Equations of Second Order. Grundlehren Math Wiss, Vol 224. Berlin: Springer-Verlag, 1983

    MATH  Google Scholar 

  29. 29.

    Golubov B. Boundedness of the Hardy and the Hardy-Littlewood operators in the spaces ReH1 and BMO. Mat Sb, 1997, 188: 93–106

    MathSciNet  Article  Google Scholar 

  30. 30.

    Hardy G H, Littlewood J E, Pólya G. Inequalities. London: Cambridge Univ Press, 1934

    MATH  Google Scholar 

  31. 31.

    Harboure E, Salinas O, Viviani B. Reverse Hölder classes in the Orlicz-spaces setting. Studia Math, 1998, 130: 245–261

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Iwaniec T, Sbordone C. Riesz transforms and elliptic PDEs with VMO coefficients. J Anal Math, 1998, 74: 183–212

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Janson S. Mean oscillation and commutators of singular integral operators. Ark Mat, 1978, 16: 263–270

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Komori Y. Notes on commutators of Hardy operators. Int J Pure Appl Math, 2003, 7: 329–334

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Krantz S, Li S Y. Boundedness and compactness of integral operators on spaces of homogeneous type and applications II. J Math Anal Appl, 2001, 258: 642–657

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Lemarioé-Rieusset P. The Navier-Stokes equations in the critical Morrey-Campanato space. Rev Mat Iberoam, 2007, 23: 897–930

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Lu G Z. Embedding theorems on Campanato-Morrey spaces for degenerate vector fields and applications. C R Acad Sci Paris Sér I Math, 1995, 320: 429–434

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Lu S Z, Yang D C. The central BMO spaces and Littlewood-Paley operators. Approx Theory Appl, 1995, 11: 72–94

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Lu S Z, Yan D Y, Zhao F Y. Sharp bounds for Hardy type operators on higher-dimensional product spaces. J Inequal Appl, 2013, 148: 1–11

    MathSciNet  Google Scholar 

  40. 40.

    Long S C, Wang J. Commutators of Hardy operators. J Math Anal Appl, 2002, 274: 626–644

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

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

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Palagachev D, Softova L. Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s. Potential Anal, 2004, 20: 237–263

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Paluszynski M. Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ Math J, 1995, 44: 1–17

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Sawyer E. Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans Amer Math Soc, 1984, 1: 329–337

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Shi S G, Fu Z W, Lu S Z. On the compactness of commutators of Hardy operators. Pacific J Math, 2020, 307: 239–256

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Shi S G, Lu S Z. Some characterizations of Campanato spaces via commutators on Morrey spaces. Pacific J Math, 2013, 264: 221–234

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Shi S G, Lu S Z. A characterization of Campanato space via commutator of fractional integral. J Math Anal Appl, 2014, 419: 123–137

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Shi S G, Lu S Z. Characterization of the central Campanato space via the commutator operator of Hardy type. J Math Anal Appl, 2015, 429: 713–732

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Monographs in Harmonic Analysis, III. Princeton Math Ser, 43. Princeton: Princeton Univ Press, 1993

    MATH  Google Scholar 

  50. 50.

    Stein E M, Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Monographs in Harmonic Analysis, I. Princeton Math Ser, 32. Princeton: Princeton Univ Press, 1971

    MATH  Google Scholar 

  51. 51.

    Uchiyama A. On the compactness of operators of Hankel type. Tohoku Math J, 1978, 30: 163–171

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Wang S M, Lu S Z, Yan D Y, Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces. Sci China Math, 2012, 55(12): 2469–2480

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Wu Q Y, Fu Z W. Weighted p-adic Hardy operators and their commutators on p-adic central Morrey spaces. Bull Malays Math Sci Soc, 2017, 40: 635–654

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Wu Q Y, Mi L, Fu Z W. Boundedness of p-adic Hardy operators and their commutators on p-adic central Morrey and BMO spaces. J Funct Spaces, 2013, 2013: 1–10

    MATH  Google Scholar 

  55. 55.

    Yang D C, Yang D Y, Zhou Y. Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators. Nagoya Math J, 2010, 198: 77–119

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Yuan W, Sickel W, Yang D C. Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Math, Vol 2005. Berlin: Springer-Verlag, 2010

    MATH  Book  Google Scholar 

  57. 57.

    Zhao F Y, Lu S Z. A characterization of λ-central BMO space. Front Math China, 2013, 8: 229–238

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

The results of this paper benefit from the author’s cooperation with Professors Dunyan Yan, Zunwei Fu, Fayou Zhao, and Shaoguang Shi in recent decades. The author thanks the anonymous referees cordially for their valuable suggestions on this paper. This work was partially supported by the National Natural Science Foundation of China (Grant No. 11771195) and the Key Laboratory of Mathematics and Complex System of Beijing Normal University.

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Correspondence to Shanzhen Lu.

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Lu, S. Function characterizations via commutators of Hardy operator. Front. Math. China 16, 1–12 (2021). https://doi.org/10.1007/s11464-021-0894-9

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Keywords

  • Hardy operator
  • commutator
  • central function space

MSC2020

  • 42B20
  • 42B25