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GRAND LEBESGUE SPACES ON QUASI-METRIC MEASURE SPACES OF INFINITE MEASURE

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Abstract

Grand Lebesgue spaces, well-known on sets of finite measure, are studied on general measure spaces \((X,\mu )\) with admission of \(\mu (X)=\infty\). In this case, the grand space is defined via the so-called grandizer. It is shown that the classical Lebesgue space is embedded into the grand Lebesgue space if and only if the grandizer is in \(L^1(X)\). Embedding between grand Lebesgue spaces with different grandizers is also considered. The main results concern the boundedness of maximal, singular, and fractional operators in grand Lebesgue spaces over quasi-metric measure spaces \((X,d, \mu )\). Application to such operators over \(\mathbb S^{n-1}\) and homogeneous groups on \(\mathbb {R}^n\) is also given.

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Funding

The research of V. Guliyev was supported by grant of Cooperation Program 2532 TUBITAK - RFBR (RUSSIAN foundation for basic research) (Agreement number no. 119N455).

The research of S. Samko and S. Umarkhadzhiev was supported by RFBR and TUBITAK according to the research project No 20-51-46003.

The research of S. Umarkhadzhiev was supported by the Ministry of Science and Higher Education RF, agreement N 075-02-2023-924 dated 16.02.2023.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Baku State University, Baku, AZ, 1148, Azerbaijan

    V. Guliyev

  2. Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141, Baku, Azerbaijan

    V. Guliyev

  3. Universidade do Algarve, Departamento de Matemática, Faculdade de Ciências, Faro, 8005, Portugal

    S. Samko

  4. Kh. Ibragimov Complex Institute of Russian Academy of Science, Grozny, Russia

    S. Samko & S. Umarkhadzhiev

  5. Academy of Sciences of Chechen Republic, Grozny, Russia

    S. Umarkhadzhiev

  6. Regional Mathematical Center of Southern Federal University, Rostov-on-Don, 344006, Russia

    S. Umarkhadzhiev

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  1. V. Guliyev
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Correspondence to S. Umarkhadzhiev.

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Guliyev, V., Samko, S. & Umarkhadzhiev, S. GRAND LEBESGUE SPACES ON QUASI-METRIC MEASURE SPACES OF INFINITE MEASURE. J Math Sci 271, 568–582 (2023). https://doi.org/10.1007/s10958-023-06578-9

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