Positivity (2022) 26:53 https://doi.org/10.1007/s11117-022-00915-z Positivity Local grand Lebesgue spaces on quasi-metric measure spaces and some applications Humberto Rafeiro 1 · Stefan Samko 2,3 · Salaudin Umarkhadzhiev 3,4 Received: 10 October 2021 / Accepted: 19 April 2022 / Published online: 30 May 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 Abstract We introduce local grand Lebesgue spaces, over a quasi-metric measure space ( X , d , μ), 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, sin- gular 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. Keywords Grand Lebesgue spaces · Maximal function · Singular integrals · Riesz potential Mathematics Subject Classification Primary: 42B35; Secondary: 42B20 · 42B25 B Humberto Rafeiro rafeiro@uaeu.ac.ae Stefan Samko ssamko@ualg.pt Salaudin Umarkhadzhiev umsalaudin@gmail.com 1 Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O.Box 15551, Al Ain, United Arab Emirates 2 Universidade do Algarve, Faro, Portugal 3 Kh. Ibragimov Complex Institute of Russian Academy of Sciences, Staropromyslovskoe shosse, 21a, Grozny, Russia 364051 4 Academy of Sciences of Chechen Republic, Esambaev av., 13, Grosny, Russia 364024 123