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Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces

Received: 23 March 2019     Accepted: 22 April 2019     Published: 9 May 2019
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Abstract

In this paper, we study the boundedness of some sublinear operators with rough kernels, satisfied by most of the operators in classical harmonic analysis, on the generalized weighted grand Morrey spaces. More specifically, we show that the sublinear operators with rough kernels are bounded on these spaces under the conditions that the operators and the kernel functions satisfy some size conditions, and the operators are bounded on Lebesgue spaces. This is done by exploiting the well-known boundedness of sublinear operators with rough kernels on Lebesgue spaces, a more explicit decomposition of the generalized weighted grand Morrey spaces and the good properties of the weight functions and the kernel functions. Through combining some properties of Ap weight with the relevant lemmas of operators with rough kernel, we obtain the boundedness for sublinear operators with rough kernels on weighted grand morrey spaces. Furthermore, using the equivalent norm and the properties of BMO functions, an application of the boundedness of the sublinear operators with rough kernels to the corresponding commutators formed by certain operators and BMO functions are also considered. And the boundedness of commutator is obtained by the lemma of function BMO.

Published in Pure and Applied Mathematics Journal (Volume 8, Issue 1)
DOI 10.11648/j.pamj.20190801.13
Page(s) 18-29
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), 2019. Published by Science Publishing Group

Keywords

Weighted Grand Morrey Space, Sublinear Operator, Rough Kernel, Commutator

References
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  • APA Style

    Junmei Wang. (2019). Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces. Pure and Applied Mathematics Journal, 8(1), 18-29. https://doi.org/10.11648/j.pamj.20190801.13

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    ACS Style

    Junmei Wang. Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces. Pure Appl. Math. J. 2019, 8(1), 18-29. doi: 10.11648/j.pamj.20190801.13

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    AMA Style

    Junmei Wang. Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces. Pure Appl Math J. 2019;8(1):18-29. doi: 10.11648/j.pamj.20190801.13

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  • @article{10.11648/j.pamj.20190801.13,
      author = {Junmei Wang},
      title = {Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces},
      journal = {Pure and Applied Mathematics Journal},
      volume = {8},
      number = {1},
      pages = {18-29},
      doi = {10.11648/j.pamj.20190801.13},
      url = {https://doi.org/10.11648/j.pamj.20190801.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20190801.13},
      abstract = {In this paper, we study the boundedness of some sublinear operators with rough kernels, satisfied by most of the operators in classical harmonic analysis, on the generalized weighted grand Morrey spaces. More specifically, we show that the sublinear operators with rough kernels are bounded on these spaces under the conditions that the operators and the kernel functions satisfy some size conditions, and the operators are bounded on Lebesgue spaces. This is done by exploiting the well-known boundedness of sublinear operators with rough kernels on Lebesgue spaces, a more explicit decomposition of the generalized weighted grand Morrey spaces and the good properties of the weight functions and the kernel functions. Through combining some properties of Ap weight with the relevant lemmas of operators with rough kernel, we obtain the boundedness for sublinear operators with rough kernels on weighted grand morrey spaces. Furthermore, using the equivalent norm and the properties of BMO functions, an application of the boundedness of the sublinear operators with rough kernels to the corresponding commutators formed by certain operators and BMO functions are also considered. And the boundedness of commutator is obtained by the lemma of function BMO.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Boundedness for Sublinear Operators with Rough Kernels on Weighted Grand Morrey Spaces
    AU  - Junmei Wang
    Y1  - 2019/05/09
    PY  - 2019
    N1  - https://doi.org/10.11648/j.pamj.20190801.13
    DO  - 10.11648/j.pamj.20190801.13
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
    SP  - 18
    EP  - 29
    PB  - Science Publishing Group
    SN  - 2326-9812
    UR  - https://doi.org/10.11648/j.pamj.20190801.13
    AB  - In this paper, we study the boundedness of some sublinear operators with rough kernels, satisfied by most of the operators in classical harmonic analysis, on the generalized weighted grand Morrey spaces. More specifically, we show that the sublinear operators with rough kernels are bounded on these spaces under the conditions that the operators and the kernel functions satisfy some size conditions, and the operators are bounded on Lebesgue spaces. This is done by exploiting the well-known boundedness of sublinear operators with rough kernels on Lebesgue spaces, a more explicit decomposition of the generalized weighted grand Morrey spaces and the good properties of the weight functions and the kernel functions. Through combining some properties of Ap weight with the relevant lemmas of operators with rough kernel, we obtain the boundedness for sublinear operators with rough kernels on weighted grand morrey spaces. Furthermore, using the equivalent norm and the properties of BMO functions, an application of the boundedness of the sublinear operators with rough kernels to the corresponding commutators formed by certain operators and BMO functions are also considered. And the boundedness of commutator is obtained by the lemma of function BMO.
    VL  - 8
    IS  - 1
    ER  - 

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Author Information
  • School of Mathematics and Statistics, Shandong Normal University, Jinan, China

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