Wiener–Luxemburg amalgam 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 not 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–Pólya 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