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Local grand Lebesgue spaces on quasi-metric measure spaces and some applications

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Abstract

We introduce local grand Lebesgue spaces, over a quasi-metric measure space \( ( X,d, \mu ) \), where the Lebesgue space is “aggrandized” not everywhere but only at a given closed set F of measure zero. We show that such spaces coincide for different choices of aggrandizers if their Matuszewska–Orlicz indices are positive. Within the framework of such local grand Lebesgue spaces, we study the maximal operator, singular operators with standard kernel, and potential type operators. Finally, we give an application to Dirichlet problem for the Poisson equation, taking F as the boundary of the domain.

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Acknowledgements

H. Rafeiro was supported by a Research Start-up Grant of United Arab Emirates University via Grant No. G00002994. The research of S. Samko was supported by Russian Foundation for Basic Research under the grant 19-01-00223 and TUBITAK and Russian Foundation for Basic research under the grant 20-51-46003. The research of S. Umarkhadzhiev was supported by TUBITAK and Russian Foundation for Basic Research under the grant 20-51-46003.

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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

    Humberto Rafeiro

  2. Universidade do Algarve, Faro, Portugal

    Stefan Samko

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

    Stefan Samko & Salaudin Umarkhadzhiev

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

    Salaudin Umarkhadzhiev

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  1. Humberto Rafeiro
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  2. Stefan Samko
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  3. Salaudin Umarkhadzhiev
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Correspondence to Humberto Rafeiro.

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Rafeiro, H., Samko, S. & Umarkhadzhiev, S. Local grand Lebesgue spaces on quasi-metric measure spaces and some applications. Positivity 26, 53 (2022). https://doi.org/10.1007/s11117-022-00915-z

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