On weighted Ces\`{a}ro function spaces

TL;DR

This paper advances the understanding of weighted Cesàro and Copson function spaces by developing new discretization techniques to characterize their structures and embeddings, addressing longstanding open problems in functional analysis.

Contribution

It introduces a novel discretization approach to characterize weighted Cesàro and Copson spaces and their embeddings, solving several open problems in the field.

Findings

Characterization of embeddings between weighted Cesàro and Copson spaces

Development of discretization techniques for these spaces

Insights into pointwise multipliers and associate 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.