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

Tom 260 / 2021

Tomasz Kiwerski, Paweł Kolwicz, Lech Maligranda Studia Mathematica 260 (2021), 45-89 MSC: Primary 46E30; Secondary 46B20, 46B42. DOI: 10.4064/sm200211-8-9 Opublikowany online: 18 February 2021

Streszczenie

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.

Autorzy

  • Tomasz KiwerskiInstitute of Mathematics
    Poznan University of Technology
    Piotrowo 3A
    60-965 Poznań, Poland
    e-mail
  • Paweł KolwiczInstitute of Mathematics
    Poznan University of Technology
    Piotrowo 3A
    60-965 Poznań, Poland
    e-mail
  • Lech MaligrandaDepartment of Engineering Sciences and Mathematics
    Luleå University of Technology
    SE-971 87 Luleå, Sweden
    and
    Institute of Mathematics
    Poznan University of Technology
    Piotrowo 3A
    60-965 Poznań, Poland
    e-mail
    e-mail

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