Local approach to order continuity in Cesàro function spaces

Abstract

The goal of this paper is to present a complete characterisation 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 $(CX{)}_a=C(X_a)$. We also find simple equivalent condition for this equality which in the case of $I=[0,1]$ comes to $X\ne L^{\infty }$. 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.

18 pages

Cited by in corpus (3)

• Isomorphic and isometric structure of the optimal domains for Hardy-type operators
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• Isometric copies of $l^{\infty }$ in Cesàro-Orlicz function spaces