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Grand Lebesgue Spaces with Mixed Local and Global Aggrandization and the Maximal and Singular Operators

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Abstract

The approach to “locally” aggrandize Lebesgue spaces, previously suggested by the authors and based on the notion of “aggrandizer”, is combined with the usual “global” aggrandization. We study properties of such spaces including embeddings, dependence of the choice of the aggrandizer and, in particular, we discuss the question when these spaces are not new, coinciding with globally aggrandized spaces, and when they proved to be new. We study the boundedness of the maximal, singular, and maximal singular operators in the introduced spaces.

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Authors and Affiliations

  1. Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates

    H. Rafeiro

  2. Universidade do Algarve, Faro, Portugal

    S. Samko

  3. Kh. Ibragimov Complex Institute of Russian Academy of Sciences, Staropromyslovskoe shosse, 21a, 364051, Grozny, Russia

    S. Samko & S. Umarkhadzhiev

  4. Academy of Sciences of Chechen Republic, Esambaev av., 13, 364024, Grozny, Russia

    S. Umarkhadzhiev

Authors
  1. H. Rafeiro
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  2. S. Samko
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  3. S. Umarkhadzhiev
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Correspondence to H. Rafeiro.

Additional information

Dedicated to Professor Oleg Besov on the occasion of his 90th birthday

The research of S. Samko and S. Umarkhadzhiev was supported by TUBITAK and Russian Foundation for Basic Research under the grant No. 20-51-46003.

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Rafeiro, H., Samko, S. & Umarkhadzhiev, S. Grand Lebesgue Spaces with Mixed Local and Global Aggrandization and the Maximal and Singular Operators. Anal Math 49, 1087–1106 (2023). https://doi.org/10.1007/s10476-023-0243-1

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  • DOI: https://doi.org/10.1007/s10476-023-0243-1

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