Local grand variable exponent Lebesgue spaces

Abstract

We introduce local grand variable exponent Lebesgue spaces, where the variable exponent Lebesgue space is “aggrandized” only at a given closed set $F$ of measure zero. It is shown that, for different aggrandizers with positive Matuszewska–Orlicz indices, the corresponding local grand variable exponent Lebesgue spaces coincide. We show that the maximal operator, singular operators, and maximal singular operators are bounded in such spaces. Lastly, an application to a Dirichlet problem for the Poisson equation, where $F$ may be chosen as the boundary of the domain, is provided within the framework of such local grand spaces.