Abstract Cesàro spaces: integral representations (Q275280)

Consider a rearrangement invariant (r.i.) Banach function space \(X\) and the Cesàro operator NEWLINE\[NEWLINE \mathcal C: f \mapsto \mathcal C(f)(x):= \frac{1}{x} \int_0^x f(t)\, dt, NEWLINE\]NEWLINE where \(f\) is an integrable function. Then the space \([\mathcal C,X]\) is defined as the function space of (classes of) functions NEWLINE\[NEWLINE [\mathcal C,X] := \big\{ f: \mathcal C(|f|) \in X \big\} NEWLINE\]NEWLINE with the norm \(\| f\|_{[\mathcal C,X]} := \| \mathcal C(|f|) \|_X\). The spaces \(\mathrm{Ces}_p=[\mathcal C,L^p]\) have been studied recently, and many of their properties are known. However, this is not the case when \(L^p\) is substituted by another rearrangement invariant space \(X\). In this paper, techniques of vector measures and vector integration are used to describe these spaces. In consequence, useful information is obtained from its description as a space of integrable functions with respect to the vector measure associated to the Cesàro operator. For example, Proposition 3.1 states that if \(X\) has order continuous norm, then \([\mathcal C,X]\) coincides with \(L^1\) of the Cesàro vector measure \(m_X\), and so is also order continuous. Some negative properties are also given. Theorem 3.3 establishes that, if \(X \neq L^\infty\) is any r.i. space, then \(L^1(m_X)\) is not reflexive and so \([\mathcal C,X]\) is not reflexive and is not r.i., and the same happens with its order continuous part. In fact, it is shown in Proposition 3.5 that in this case the space is order isomorphic to an AL-space.NEWLINENEWLINEThese results allow to show particular applications for \(X\) being a relevant r.i. Banach function space, as Lorentz and Marcinkiewicz spaces, providing concrete representations of the corresponding spaces \([\mathcal C,X]\), for example as weighted \(L^1\)-spaces.NEWLINENEWLINEThe paper contains a lot of information about the description and main properties of the spaces \([\mathcal C, X]\) that may help the interested researcher in applications of these Cesàro spaces.