On weighted Cesàro function spaces
Amiran Gogatishvili, Tuğçe Ünver
TL;DR
The paper addresses the problem of characterizing embeddings between weighted Cesàro and Copson function spaces, which extend weighted Lebesgue norms through local and global iterated operators. It uses a discretization technique inspired by Grosse-Erdmann to avoid duality constraints and obtain complete, unrestricted embedding characterizations. The main contributions are explicit, parameter-free criteria for embeddings such as $\operatorname{Ces}_{p_1,q_1}(u_1,v_1)\hookrightarrow \operatorname{Ces}_{p_2,q_2}(u_2,v_2)$ and analogous Copson/Cesàro relations, expressed via constants $C_1$ through $C_7$ built from weights. These results illuminate the structure of multipliers, associate spaces, and Morrey-type connections, providing a unified approach to weighted Hardy-type inequalities in this setting.
Abstract
The main objective of this paper is to provide a comprehensive demonstration of recent results regarding the structures of the weighted Cesàro 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 Cesàro and Copson function spaces. The characterization of these embeddings can be used to tackle the problems of characterizing pointwise multipliers between weighted Cesàro and Copson function spaces, the characterizations of the associate spaces of Cesàro (Copson) function spaces, as well as the relations between local Morrey-type spaces.