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 $L^1$, namely, they contain “in the middle” a complemented copy of $L^1[0,1]$ and an asymptotically isometric copy of $\ell ^1$, and can also be renormed to contain an isometric copy of $L^1[0,1]$. Moreover, 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 Cesàro construction $X \mapsto CX$ 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.