New examples of $K$-monotone weighted Banach couples
Volume 218 / 2013
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.
