The dual of Cesàro function spaces. (English) Zbl 0647.46033

In this note X denotes a function space of all measurable functions defined on \((0,\infty)\). The authors investigate the Cesàro function space \[ CES_ p=\{f\in X: [\int^{\infty}_{0}(1/x\int^{x}_{0}| f(t)| dt)^ p dx]^{1/p}<\infty,\quad 1<p<\infty \} \] and the Köthe dual of \(CES_ p\) as \[ \{g\in X: \int^{\infty}_{0}f(x)g(x)dx\text{ exists for every } f\in CES_ p\}. \] The main result of the note is the very important theorem (Theorem 3) in which the authors characterize the Köthe dual of \(CES_ p\) in terms of Cesàro sequence space and reverse Cesàro sequence space.
Reviewer: A.Waszak


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)