Core Decreasing Functions

TL;DR

This paper introduces core decreasing functions and down spaces in Banach function spaces, extending known constructions to a general setting and characterizing their duals and interpolation properties.

Contribution

It defines core decreasing functions and down spaces, extends the least core decreasing majorant and level function constructions, and characterizes duals and interpolation spaces for these constructs.

Findings

Down spaces of L^1 and L^∞ form an exact Calderón couple with divisibility constant 1.

Complete description of exact interpolation spaces for these couples using level functions.

Down spaces of u.r.i. spaces are exactly the interpolation spaces with the Fatou property.

Abstract

Given a measure space and a totally ordered ordered collection of measurable sets, called an ordered core, the notion of a core decreasing function is introduced and used to define the down space of a Banach function space. This is done using a variant of the K\"othe dual restricted to core decreasing functions. To study down spaces, the least core decreasing majorant construction and the level function construction, already known for functions on the real line, are extended to this general setting. These are used to give concrete descriptions of the duals of the down spaces and, in the case of universally rearrangement invariant (u.r.i.) spaces, of the down spaces themselves. The down spaces of $L^1$ and $L^\infty$ are shown to form an exact Calder\'on couple with divisibility constant $1$; a complete description of the exact interpolation spaces for the couple is given in terms of level functions; and the down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property. The dual couple is also an exact Calder\'on couple with divisibility constant $1$; a complete description of the exact interpolation spaces for the couple is given in terms of least core decreasing majorants; and the duals of down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property.