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

TL;DR

This paper investigates the absolute continuity of (quasi)norms in rearrangement-invariant spaces, providing new tools and characterizations, including a detailed analysis of weak Marcinkiewicz spaces.

Contribution

It introduces a novel construction of a representation quasinorm and develops new tools to analyze absolute continuity in rearrangement-invariant spaces.

Findings

Characterization of functions with absolutely continuous quasinorms in weak Marcinkiewicz spaces

Explicit construction of a suitable representation quasinorm

Development of new analytical tools for rearrangement-invariant spaces

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{\' o}lya 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.