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

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