On the sums of rearrangement-invariant quasi-Banach function spaces and their relationship to amalgams

TL;DR

This paper investigates the properties of sums of rearrangement-invariant quasi-Banach function spaces, establishing conditions under which these sums retain rearrangement-invariance and can be characterized as Wiener--Luxemburg amalgams.

Contribution

It provides new insights into when sums of such spaces are equivalent to rearrangement-invariant quasinorms and characterizes these sums as Wiener--Luxemburg amalgams.

Findings

Quasinorms of sums are often equivalent to rearrangement-invariant quasinorms.

Sums can be characterized as Wiener--Luxemburg amalgams under certain conditions.

Provides weaker Luxemburg-type representations for sums.

Abstract

In this paper we consider the properties of sums of rearrangement-invariant quasi-Banach function spaces, with the focus being on rearrangement-invariance and the Fatou property. In our first main result, we show that the quasinorm of the sum is in many cases equivalent to a rearrangement-invariant quasinorm by providing a weaker version of the Luxemburg-type representation. In our second main result, we show that the sum can be in some cases characterised as a Wiener--Luxemburg amalgam of the two constituent spaces, thus providing a sufficient condition for the sum being a rearrangement-invariant quasi-Banach function space.