Wiener-Luxemburg amalgam spaces
Dalimil Peša
TL;DR
This work introduces Wiener--Luxemburg amalgam spaces $WL(A,B)$ to overcome limitations of classical Wiener amalgams in the framework of rearrangement-invariant Banach and quasi-Banach function spaces. It provides a comprehensive structural theory, detailing normability, embeddings, and associate spaces, and extends the framework to quasi-Banach spaces via integrable associate constructions, enabling a negative answer to the Hardy--Littlewood--Pólya principle for all rearrangement-invariant quasi-Banach norms. The analysis clarifies how local and global components interact to govern embeddings and duality, and it includes concrete results such as $WL(A,B)$ being rearrangement-invariant and sandwiched between $A\cap B$ and $A+B$ under suitable conditions. Together with counterexamples and integrable-duality concepts, the paper offers a robust toolkit for studying global-local behavior in RI spaces, with implications for Sobolev-type embeddings and interpolation theory in this generalized setting.
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-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 invariance.