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Boundedness of Littlewood-Paley Operators in Variable Morrey Spaces

Received: 14 February 2022     Accepted: 7 March 2022     Published: 17 March 2022
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Abstract

In this paper, the authors prove norm inequalities for the intrinsic square functions and commutators generated by this class operator and BMO function in variable Morrey spaces. This implies that the same norm inequalities for the Lusin area integrals, the Littlewood-Paley operators and the continuous square functions. As application, we get the boundedness for convolution Calderón-Zygmund operators in generalized Morrey spaces.

Published in American Journal of Applied Mathematics (Volume 10, Issue 2)
DOI 10.11648/j.ajam.20221002.11
Page(s) 15-28
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2022. Published by Science Publishing Group

Keywords

Littlewood-Paley Operators, Singular Integrals, Morrey Spaces, Commutator

References
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[2] L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents: Lecture Notes in Math., Vol.2017, Spring-Verlag, 2011.
[3] L. Diening, Maximal functions on generalized Lp(x) spaces, Math. Inequal Appl. 7 (2004), 245-253.
[4] L. Pick and R˚ uˇ ziˇ cka M., An example of a space Lp(x) on which the Hardy-Littlewood maximal operator is not bouded, Expo. Math 4 (2001), 369-372.
[5] D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. Pérez, The bounedness of classical operators on variable Lpspaces, Annales Academiæ Scientarum Fennicæ Mathematica 31 (2006), 239-264.
[6] V. S. Guliyev, J. J. Hasanov and S. G. Samko, Boundedness of the maximal,potential and singular operators in generalized varibale exponent Morrey spaces, Math. Scand., 107 (2010), 285-304.
[7] V. S. Guliyev, J. J. Hasanov and S. G. Samko, Boundedness of the maximal,potential and singular operators in generalized varibale exponent Morrey spaces, Journal of Mathematical Sciences, 4 (170) (2010), 423-443.
[8] A. Almeida, J. J. Hasanov and S. G. Samko, Maximal and potential in variable exponent Morrey spaces, Georgian Math. J. 15 (2008), 195-208.
[9] S. G. Samko, Convolution type operators in Lp(x), Integral Transform. Special Funct. 7 (1998), 123-144.
[10] E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces,Math. Nachr., 166 (1994), 95-103.
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[12] P. Hästö, Local-to-global results in variable exponent spaces, Math. Res. Lett., 16 (2009), no. 2, 263¨C278.
[13] Y. Mizuta and T. Shimomura, Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent, J. Math. Soc. Japan, 60 (2008), 583-602.
[14] F. Chiarenza and M. Frasca, Morrey and Hardy- Littlewood maximal function, Rend. Mat., 7 (1987), 273- 279.
[15] J. M. Wilson, Weighted norm inequalities for the continuous square functions, Tran. Amer. Math., 314 (1989), 661-692.
[16] J. M. Wilson, The intrinsic square function, Rev. MAT. Iberoam., 23 (2007), 771-791.
[17] J. M. Wilson, Weighted Littlewood-Paley theory and exponential-square integrablity, Lecture Notes in Math., Vol.1924, Spring-Verlag, 2008.
[18] A. K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singualr integras, Adv. Math., 226 (2011), 3912-3926.
[19] H. Wang, Boundedness of intrinsic square functions on generalized Morrey spaces, Georgian Math. J., 3 (21) (2014), 351-367.
[20] H. Wang, Intrinsic square functions on weighted Morrey spaces, J. Math. Anal. Appl., 396 (2012), 302-314.
[21] D. E. Edmunds, V. M. Kokilashvilii and A. Meskhi, On the boundedness and compactness of weighted Hardy operators in Lp(x), Georg. Math. J, 12 (1) (2005), 27- 44.
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  • APA Style

    Panwang Wang, Zongguang Liu. (2022). Boundedness of Littlewood-Paley Operators in Variable Morrey Spaces. American Journal of Applied Mathematics, 10(2), 15-28. https://doi.org/10.11648/j.ajam.20221002.11

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    ACS Style

    Panwang Wang; Zongguang Liu. Boundedness of Littlewood-Paley Operators in Variable Morrey Spaces. Am. J. Appl. Math. 2022, 10(2), 15-28. doi: 10.11648/j.ajam.20221002.11

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    AMA Style

    Panwang Wang, Zongguang Liu. Boundedness of Littlewood-Paley Operators in Variable Morrey Spaces. Am J Appl Math. 2022;10(2):15-28. doi: 10.11648/j.ajam.20221002.11

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  • @article{10.11648/j.ajam.20221002.11,
      author = {Panwang Wang and Zongguang Liu},
      title = {Boundedness of Littlewood-Paley Operators in Variable Morrey Spaces},
      journal = {American Journal of Applied Mathematics},
      volume = {10},
      number = {2},
      pages = {15-28},
      doi = {10.11648/j.ajam.20221002.11},
      url = {https://doi.org/10.11648/j.ajam.20221002.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221002.11},
      abstract = {In this paper, the authors prove norm inequalities for the intrinsic square functions and commutators generated by this class operator and BMO function in variable Morrey spaces. This implies that the same norm inequalities for the Lusin area integrals, the Littlewood-Paley operators and the continuous square functions. As application, we get the boundedness for convolution Calderón-Zygmund operators in generalized Morrey spaces.},
     year = {2022}
    }
    

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    T1  - Boundedness of Littlewood-Paley Operators in Variable Morrey Spaces
    AU  - Panwang Wang
    AU  - Zongguang Liu
    Y1  - 2022/03/17
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    N1  - https://doi.org/10.11648/j.ajam.20221002.11
    DO  - 10.11648/j.ajam.20221002.11
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    EP  - 28
    PB  - Science Publishing Group
    SN  - 2330-006X
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    AB  - In this paper, the authors prove norm inequalities for the intrinsic square functions and commutators generated by this class operator and BMO function in variable Morrey spaces. This implies that the same norm inequalities for the Lusin area integrals, the Littlewood-Paley operators and the continuous square functions. As application, we get the boundedness for convolution Calderón-Zygmund operators in generalized Morrey spaces.
    VL  - 10
    IS  - 2
    ER  - 

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Author Information
  • Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, P. R. China

  • Department of Mathematics, China University of Mining and Technology, Beijing, P. R. China

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