Understanding Function Spaces, Morrey-Campanato Spaces and Some Generalizations

The theory of Campanato-Morrey spaces, along with their applications and generalizations, has evolved significantly over the last few decades. These spaces now play a fundamental role in modern Function and Harmonic Analysis, as well as in the regularity theory of Partial Differential Equations, alongside classical H\"older and Sobolev spaces. In his landmark paper "On the solutions of quasi-linear elliptic partial differential equations", Morrey introduced a novel approach to studying the local Holder regularity of solutions to second-order elliptic operators. This method involved estimating the growth of an integral function of the gradient on a ball in terms of the power of the same ball radius. Although Morrey did not explicitly introduce function spaces, his paper is widely regarded as the foundation for the theory of Morrey spaces and their numerous generalizations. In this chapter, we present a brief overview of the key properties, selected embedding results, and the most significant generalizations of the aforementioned function spaces.