Absolute continuity of the (quasi)norm in rearrangement-invariant spaces

2024/12/18 by Dalimil Peša, Peša, Dalimil · 1 citation
Mathematics · #46A16 #46E30 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2412.13903

openalex publication_date 2024/12/18 · openalex created_date 2024/12/21 · openalex updated_date 2026/07/30

Abstract

This paper explores the interactions of absolute continuity of the (quasi)norm with the concepts that are fundamental in the theory of rearrangement-invariant (quasi-)Banach function spaces, such as the Luxemburg representation or the Hardy--Littlewood--P' olya relation. In order to prove our main results, we give an explicit construction of a particularly suitable representation quasinorm (which is not necessarily unique) and develop several new tools that we believe to be of independent interest. As an application of our results, we characterise the subspace of functions having absolutely continuous quasinorms in weak Marcinkiewicz spaces.

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• On the sums of rearrangement-invariant quasi-Banach function spaces and their relationship to amalgams

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