On the sums of rearrangement-invariant quasi-Banach function spaces and their relationship to amalgams
Dalimil Peša
TL;DR
This work studies sums of rearrangement-invariant quasi-Banach function spaces, focusing on whether RI structure and Fatou properties transfer to $A+B$ and how intersections behave. It develops a Luxemburg-type representation via canonical representation spaces to show that, in non-atomic settings, $\|f\|_{A+B} \asymp \|f^*\|_{\overline{A}+\overline{B}}$, and that $\|f^*\|_{\overline{A}+\overline{B}}$ preserves key RI-quasinorm properties, with atomic cases requiring Hardy–Littlewood–Pólya conditions. The paper then relates these sums to Wiener–Luxemburg amalgams $WL(A,B)$, proving embeddings $A\cap B \hookrightarrow WL(A,B) \hookrightarrow A+B$ and, under suitable embeddings, equalities up to quasinorms; it also connects the amalgam framework to canonical representation spaces via $\overline{WL(A,B)}=WL(\overline{A},\overline{B})$ and $\overline{A\cap B}=\overline{A}\cap\overline{B}$. Overall, the results provide representation- and amalgam-based tools to transfer RI and Fatou properties to sums of non-classical spaces and clarify when these sums align with amalgams across non-atomic and atomic regimes.
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.