Local approach to order continuity in Cesàro function spaces

Abstract

Abstract. The goal of this paper is to present a complete characterization of points of order continuity in abstract Cesàro function spaces CX for X being a symmetric function space. Under some additional assumptions mentioned result takes the form (CX)a = C(Xa). We also find simple equivalent condition for this equality which in the case of I = [0, 1] comes to X = L ∞ . Furthermore, we prove that X is order continuous if and only if CX is, under assumption that the Cesàro operator is bounded on X. This result is applied to particular spaces, namely: Cesàro-Orlicz function spaces, Cesàro-Lorentz function spaces and Cesàro-Marcinkiewicz function spaces to get criteria for OC-points.

Isometric Copies of $$\varvec{l^\infty }$$ l ∞ in Cesàro–Orlicz Function Spaces

“…Finally, the last part of this lemma follows easily from more general result concerning symmetric function spaces, see [22,Lemma 14].…”

“…Now, we will prove that supp (Ces ϕ ) a = I. This fact is true even if we take symmetric function space X instead of Orlicz function space L ϕ , i.e., supp(CX) a = I for any symmetric function spaces X on I, see [22,Lemma 10]. However, we present the details of the proof because in the case of X = L ϕ we can use a shorter and easier argument.…”

Quotients, $\ell_\infty$ and abstract Cesàro spaces

“…Put A = (0, a) ∪ (b, ∞) and A c = (a, b). In view of Theorem 3.5, we have the equality [KT17,Lemma 8] in the case of function spaces; however, simple modification of this argument works in the sequence case as well). Consequently, taking h = f * χ A , we have…”

“…The proof of the above theorem makes strong use of Theorem 3.5 from §3 and Theorem 16 from [KT17]. Note that Theorem 16 from [KT17] has been proved for rearrangement invariant function, but not sequence, spaces. For the missing proof, we refer to Appendix.…”

“…Finally, at the end of this paper, we include Appendix, which was created for the purposes of §4 (and, at the same time, it complements the results obtained by the first and last author in [KT17]). It contains a fairly satisfactory description of the ideal of all order continuous functions in the Cesàro sequence spaces CX expressed in the language of the ideal X a .…”