New examples of $K$-monotone weighted Banach couples

Sergey V. Astashkin¹ ; Lech Maligranda² ; Konstantin E. Tikhomirov³

¹Department of Mathematics and Mechanics Samara State University Acad. Pavlova 1 443011 Samara, Russia
²Department of Engineering Sciences and Mathematics Luleå University of Technology SE-971 87 Luleå, Sweden
³Department of Mathematical and Statistical Sciences University of Alberta Edmonton, AB T6G2G1, Canada

Studia Mathematica, Tome 218 (2013) no. 1, pp. 55-88

Résumé

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.

Détail

DOI : 10.4064/sm218-1-4

Keywords: examples k monotone couples type where symmetric space weight presented based property w decomposability symmetric space weight changes sufficiently fast symmetric spaces non trivial boyd indices banach couple k monotone belong class ultrasymmetric orlicz spaces addition fundamental function infty time banach couple may k monotone non trivial ultrasymmetric each cases where lorentz marcinkiewicz orlicz space conditions which guarantee k monotone