On the dual of Ces\`aro function space

TL;DR

This paper characterizes the dual space of Cesàro function spaces using an isometric representation, introduces the concept of essential -concave majorant, and explores geometric properties of these spaces.

Contribution

It provides a new isometric representation of the dual space of Cesàro function spaces and introduces the notion of essential -concave majorant.

Findings

Every slice of the unit ball has diameter 2

Cesro spaces lack the Radon-Nikodym property

Cesro spaces are not locally uniformly convex or dual spaces

Abstract

The goal of this paper is to present an isometric representation of the dual space to Ces\`aro function space $C_{p,w}$, $1<p<\infty$, induced by arbitrary positive weight function $w$ on interval $(0,l)$ where $0<l\leqslant\infty$. For this purpose given a strictly decreasing nonnegative function $\Psi$ on $(0,l)$, the notion of essential $\Psi$-concave majorant $\hat f$ of a measurable function $f$ is introduced and investigated. As applications it is shown that every slice of the unit ball of the Ces\`aro function space has diameter 2. Consequently Ces\`aro function spaces do not have the Radon-Nikodym property, are not locally uniformly convex and they are not dual spaces.