New examples of K-monotone weighted Banach couples

TL;DR

This paper explores new examples of K-monotone couples of symmetric spaces and weights, revealing conditions under which these couples are K-monotone, especially focusing on ultrasymmetric Orlicz spaces and specific space types.

Contribution

It provides new examples and characterizations of K-monotone couples involving symmetric spaces and weights, including conditions for ultrasymmetric Orlicz, Lorentz, and Marcinkiewicz spaces.

Findings

K-monotone couples are characterized for ultrasymmetric Orlicz spaces.

Conditions for K-monotonicity are established for Lorentz, Marcinkiewicz, and Orlicz spaces.

Fast-changing weights influence the class of symmetric spaces forming K-monotone couples.

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 the 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 have found conditions which guarantee that (X, X(w)) is K-monotone.