### Abstract

Some new examples of K-monotone couples of the type (X;X(ω)), where X is a symmetric space on [0; 1] and ω is a weight on [0; 1], are presented. Based on the property of ω-decomposability of a symmetric space we show that, if a weight ω changes sufficiently fast, all symmetric spaces X with non-trivial Boyd indices such that the Banach couple (X;X(ω)) 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 ⋯ [1;∞], then X = L^{p}. At the same time a Banach couple (X;X(ω)) 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(ω)) is K-monotone.

Original language | English (US) |
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Pages (from-to) | 55-88 |

Number of pages | 34 |

Journal | Studia Mathematica |

Volume | 218 |

Issue number | 1 |

DOIs | |

State | Published - Nov 15 2013 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- K-functional
- K-method of interpolation
- K-monotone couples
- Lorentz spaces
- Marcinkiewicz spaces
- Omega;-decomposable Banach lattices
- Orlicz spaces
- Regularly varying functions
- Symmetric spaces
- Ultrasymmetric spaces
- Weighted symmetric spaces

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## Cite this

*Studia Mathematica*,

*218*(1), 55-88. https://doi.org/10.4064/sm218-1-4