ABSTRACT
In this chapter, the authors consider Morrey spaces over open sets. This includes Sobolev–Morrey spaces. Since many partial differential equations are considered over domains, it is meaningful to consider Morrey spaces over open sets Ω. They authors investigate some maximal inequalities for Morrey spaces with non-doubling measures. The authors know that Morrey spaces can describe the local regularity and the global regularity more precisely than Lebesgue spaces. Sobolev–Morrey spaces arose in the study of elliptic differential equations. Campanato considered Sobolev–Morrey spaces. Guliyev and Mustafayev considered singular integral operators and the maximal operators acting on generalized Morrey spaces in the anisotropic setting. In the general metric measure setting, the authors need to equip Morrey spaces with another parameter, which is not of importance in the Eucledean space. The Gauss measure space falls under the scope of the framework of locally doubling Morrey spaces.
