Two improvements upon the paper "Wiener--Luxemburg amalgam spaces"

Abstract: This short note seeks to present two new results that essentially improve the corresponding statements from the author's paper ``Wiener--Luxemburg amalgam spaces''. Specifically, in our first result, we show that when both the local and the global components of a pair of r.i. quasi-Banach function spaces are appropriately ordered, then both their sum and intersection can be characterised as the appropriate amalgam of said spaces, up to equivalence of quasinorms. Then, in our second result, we provide a sufficient condition for the validity of the Hardy--Littlewood--P\'{o}lya principle that is weaker than the one obtained originally. We also discuss some consequences of those results and provide further context.