Local approach to order continuity in Ces\`aro function spaces

The goal of this paper is to present a complete characterisation of points of order continuity in abstract Ces\`aro 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 $X\neq L^\infty$. Furthermore, we prove that $X$ is order continuous if and only if $CX$ is, under assumption that the Ces\`aro operator is bounded on $X$. This result is applied to particular spaces, namely: Ces\`aro-Orlicz function spaces, Ces\`aro-Lorentz function spaces and Ces\`aro-Marcinkiewicz function spaces to get criteria for OC-points.Comment: 18 page

Isomorphic and isometric structure of the optimal domains for Hardy-type operators

We investigate structure of the optimal domains for the Hardy-type operators including, for example, the classical Ces\`aro, Copson and Volterra operators as well as for some of their generalizations. We prove that, in some sense, the abstract Ces\`aro and Copson function spaces are closely related to the space $L^1$, namely, they contain "in the middle" a complemented copy of $L^1[0,1]$, asymptotically isometric copy of $\ell^1$ and also can be renormed to contain an isometric copy of $L^1[0,1]$. Moreover, the generalized Tandori function spaces are quite similar to $L^\infty$ because they contain an isometric copy of $\ell^\infty$ and can be renormed to contain an isometric copy of $L^\infty[0,1]$. Several applications to the metric fixed point theory will be given. Next, we prove that the Ces\`aro construction $X \mapsto CX$ does not commutate with the truncation operation of the measure space support. We also study whether a given property transfers between a Banach function space $X$ and the space $TX$, where $T$ is the Ces\`aro or the Copson operator. In particular, we find a large class of properties which do not lift from $TX$ into $X$ and prove that the abstract Ces\`aro and Copson function spaces are never reflexive, are not isomorphic to a dual space and do not have the Radon--Nikodym property in general.Comment: 34 pages; we changed the title and added some corrections compared to the first versio

Isomorphic and isometric structure of the optimal domains for Hardy-type operators

We investigate the structure of optimal domains for the Hardy-type operators including, for example, the classical Cesàro, Copson and Volterra operators as well as some of their generalizations. We prove that, in some sense, the abstract Cesàro and Copson function spaces are closely related to the space L1, namely, they contain “in the middle” a complemented copy of L1[0,1] and an asymptotically isometric copy of ℓ1, and can also be renormed to contain an isometric copy of L1[0,1]. Moreover, generalized Tandori function spaces are quite similar to L∞ because they contain an isometric copy of ℓ∞ and can be renormed to contain an isometric copy of L∞[0,1]. Several applications to the metric fixed point theory will be given. Next, we prove that the Cesàro construction X↦CX does not commute with the truncation operation of the measure space support. We also study whether a given property transfers between a Banach function space X and the space TX, where T is the Cesàro or the Copson operator. In particular, we find a large class of properties which do not lift from TX into X and we prove that abstract Cesàro and Copson function spaces are never reflexive, are not isomorphic to a dual space and do not have the Radon–Nikodym property in general.Validerad;2021;Nivå 2;2021-06-28 (alebob);Finansiär: Ministry of Science and Higher Education of Poland (0213/SBAD/0114)</p