Disjointly homogeneous rearrangement invariant spaces via interpolation

TL;DR

This paper characterizes p-disjointly homogeneous rearrangement invariant spaces on [0,1] using interpolation theory, identifying conditions for their existence and uniqueness based on fundamental functions and dilation indices.

Contribution

It provides a complete description of p-disjointly homogeneous r.i. spaces via interpolation methods, including existence and uniqueness results.

Findings

Existence of p-disjointly homogeneous r.i. spaces for various fundamental functions.

Uniqueness of such spaces within certain interpolation classes.

Characterization of these spaces using dilation indices and fundamental functions.

Abstract

A Banach lattice E is called p-disjointly homogeneous, 1< p< infty, when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of l_p. Employing methods from interpolation theory, we clarify which rearrangement invariant (r.i.) spaces on [0,1] are p-disjointly homogeneous. In particular, for every 1<p< infty and any increasing concave function f on [0,1], which is not equivalent neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function f. Moreover, in the class of all interpolation r.i. spaces with respect to the Banach couple of Lorentz and Marcinkiewicz spaces with the same fundamental function, dilation indices of which are non-trivial, for every 1<p< infty, there is only a unique p-disjointly homogeneous space.