On weighted Ces\`{a}ro function spaces

Abstract

The main objective of this paper is to provide a comprehensive demonstration of recent results regarding the structures of the weighted Ces\`aro and Copson function spaces. These spaces' definitions involve local and global weighted Lebesgue norms; in other words, the norms of these spaces are generated by positive sublinear operators and by weighted Lebesgue norms. The weighted Lebesgue spaces are the special cases of these spaces with a specific set of parameters. Our primary method of investigating these spaces will be the so-called discretization technique. Our technique will be the development of the approach initiated by K.G. Grosse-Erdmann, which allows us to obtain the characterization in previously unavailable situations, thereby addressing decades-old open problems. We investigate the relation (embeddings) between weighted Ces\`aro and Copson function spaces. The characterization of these embeddings can be used to tackle the problems of characterizing pointwise multipliers between weighted Ces\`aro and Copson function spaces, the characterizations of the associate spaces of Ces\`aro (Copson) function spaces, as well as the relations between local Morrey-type spaces.