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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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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DOI: https://doi.org/10.1007/s11117-022-00915-z