New examples of $K$-monotone weighted Banach couples

Volume 218 / 2013

Sergey V. Astashkin, Lech Maligranda, Konstantin E. Tikhomirov Studia Mathematica 218 (2013), 55-88 MSC: Primary 46B70, 46E30; Secondary 46B20, 46B42. DOI: 10.4064/sm218-1-4

Abstract

Some new examples of $K$-monotone couples of the type $(X, X(w))$, where $X$ is a symmetric space on $[0, 1]$ and $w$ is a weight on $[0, 1]$, are presented. Based on the property of $w$-decomposability of a symmetric space we show that, if a weight $w$ changes sufficiently fast, all symmetric spaces $X$ with non-trivial Boyd indices such that the Banach couple $(X, X(w))$ is $K$-monotone belong to the class of ultrasymmetric Orlicz spaces. If, in addition, the fundamental function of $X$ is $t^{1/p}$ for some $p \in [1, \infty ]$, then $X = L_p$. At the same time a Banach couple $(X, X(w))$ may be $K$-monotone for some non-trivial $w$ in the case when $X$ is not ultrasymmetric. In each of the cases where $X$ is a Lorentz, Marcinkiewicz or Orlicz space, we find conditions which guarantee that $(X, X(w))$ is $K$-monotone.

Authors

  • Sergey V. AstashkinDepartment of Mathematics and Mechanics
    Samara State University
    Acad. Pavlova 1
    443011 Samara, Russia
    e-mail
  • Lech MaligrandaDepartment of Engineering Sciences
    and Mathematics
    Luleå University of Technology
    SE-971 87 Luleå, Sweden
    e-mail
  • Konstantin E. TikhomirovDepartment of Mathematical and Statistical Sciences
    University of Alberta
    Edmonton, AB T6G2G1, Canada
    e-mail

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