Abstract Cesàro Spaces. Optimal Range

Abstract

Cesàro spaces are investigated from the optimal domain and optimal range point of view. There is a big difference between the cases on [0, ∞) and on [0, 1], as we can see in Theorem 1. Moreover, we present an improvement of Hardy inequality on [0, 1] which plays an important role in these considerations.*This publication has been written during scholarship period of the first author at the Luleå University of Technology, thanks to a Swedish Institute scholarschip (number 0095/2013).

Local approach to order continuity in Cesàro function spaces

“…for 0 < x ∈ I. For a Banach ideal space X on I we define an abstract Cesàro space CX = CX(I) by CX := {f ∈ L 0 (I) : C|f | ∈ X}, with the norm f CX = C|f | X (see [9], [10], [22], [23], [24]). Let us note that for nonsymmetric space X the space CX need not have a weak unit even if X has it (see [22,Example 2]), so in general supp(CX) ⊂ supp(X).…”

Isomorphic Structure of Cesàro and Tandori Spaces

“…Most commonly the classical Cesàro spaces appear as optimal domains of the Cesàro (Hardy) operator or some its versions (see [DS07], [NP10], [LM15b]). Moreover, they can coincide with the so-called down spaces introduced and investigated by Sinnamon (see [KMS07], [MS06], [Si94], [Si01], [Si07]), but having their roots in the papers of Halperin and Lorentz.…”

“…[HS73, Theorem 1]), we see that CX contains also a subspace isomorphic to C[0, 1] * . However, it is impossible, since CX is separable (by Lemma 1 in [LM15b]).…”

Embeddings between weighted Tandori and Cesàro function spaces

“…Furthermore, in 2015, Lesnik and Maligranda [16,17] began studying these spaces within an abstract framework, where they used a more general function space X instead of the weighted Lebesgue spaces. When X is a Banach space, they de-…ned Cesàro space CX, Copson space C X and Tandori space e X as the set of all measurable functions, respectively, with the following norms:…”