Local grandization of Lebesgue spaces

In this talk, we introduce and discuss 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 null set $F$. Within the framework of such spaces, we study the maximal operator, singular operators with standard kernel, and potential type operators. Furthermore, we examine an application to the Dirichlet problem for the Poisson equation, taking $F$ as the boundary of the domain. Lastly, we introduce and explore the concept of grand Lebesgue spaces with mixed local and global aggrandization.