Structure of Cesàro function spaces: a survey

Factorizations of Weighted Hardy Inequalities

“…The results of Bennett (1996) have a big impact in many parts of analysis but it seems that the corresponding results for weighted spaces are less studied. For a recent survey of results on classical Cesàro spaces see Astashkin and Maligranda (2014). In the newly, very interesting papers, Leśnik and Maligranda (2015), Kolwicz et al (2014) similar results and motivations, in an abstract, very general setting were presented.…”

Abstract Cesàro spaces: Integral representations

“…Cesàro function spaces have attracted much attention in recent times; see for example the papers [1], [2], [3] by Astashkin and Maligranda and [17], [18] by Lésnik and Maligranda and the references therein. These spaces arise when studying the behavior, in certain function spaces, of the Cesàro operator We will focus our attention on those spaces X which are rearrangement invariant (r.i.) on [0, 1].…”

“…We only consider r.i. spaces X = L ∞ because Ces ∞ = [C, L ∞ ], known as the Korenblyum-Kreȋn-Levin space, has already been thoroughly investigated; see [2], [3] and the references therein.…”

Rotundity and monotonicity properties of selected Cesàro function spaces

“…If the above inequalities are satisfied for u ∈ [u 0 , ∞) with u 0 > 0 such that ψ(u 0 ) > 0 then we say that ψ, γ are equivalent (weak equivalent) for "large arguments" and we write shortly ψ ∼ l γ (ψ w ∼ l γ ). The proof of the next fact is just an easy exercise -it is enough to apply equivalent formulation for condition 2 , see [8]. …”

“…The classical Cesàro sequence ces p and function Ces p spaces have been studied by many authors (see [1,2] also for further references). It is worth to mention that some properties are fulfilled in the sequence case and are not in function case.…”