Wiener-Luxemburg amalgam spaces

TL;DR

This paper introduces Wiener-Luxemburg amalgam spaces as a modification of Wiener amalgam spaces, studying their properties, embeddings, and associate spaces, and applies the theory to Hardy-Littlewood-Pólya principles and counterexamples.

Contribution

It proposes Wiener-Luxemburg amalgam spaces, extending the classical framework, and addresses their properties, embeddings, and applications to quasi-Banach spaces and rearrangement-invariant properties.

Findings

Wiener-Luxemburg amalgam spaces are normable and have well-defined embeddings.

Counterexamples demonstrate limitations of Wiener amalgam spaces.

The Hardy-Littlewood-Pólya principle does not hold universally for all r.i. quasi-Banach spaces.

Abstract

In this paper we introduce the concept of Wiener-Luxemburg amalgam spaces which are a modification of the more classical Wiener amalgam spaces intended to address some of the shortcomings the latter face in the context of rearrangement-invariant Banach function spaces. We introduce the Wiener-Luxemburg amalgam spaces and study their properties, including (but nor limited to) their normability, embeddings between them and their associate spaces. We also study amalgams of quasi-Banach function spaces and introduce a necessary generalisation of the concept of associate spaces. We then apply this general theory to resolve the question whether the Hardy-Littlewood-P\'{o}lya principle holds for all r.i. quasi-Banach function spaces. Finally, we illustrate the asserted shortcomings of Wiener amalgam spaces by providing counterexamples to certain properties of Banach function spaces as well as rearrangement invariance.