Absolute continuity of the (quasi)norm in rearrangement-invariant spaces
Dalimil Peša
TL;DR
This work addresses how absolute continuity of the (quasi)norm interacts with rearrangement-invariant (quasi-)Banach spaces, establishing representation-based criteria that connect AC of $X$ to AC of its representation $\overline{X}$ via the nonincreasing rearrangement $f^*$ and a carefully constructed quasinorm. The authors construct a particular representation quasinorm on $([0,\infty), \lambda)$ and prove that AC of the quasinorm is preserved under this representation, including compatibility with the Hardy-Littlewood-Pólya relation. They obtain affirmative answers to two natural questions: whether AC of the quasinorm can be characterized through the rearrangement and whether HLPR-majorization preserves AC when the dominating function has AC; and they develop tools linking these concepts that yield applications to weak Marcinkiewicz spaces, where the AC-quasinorm subspace is nontrivial. Overall, the results extend the theory of RI quasi-Banach spaces, providing concrete criteria and representation-based techniques for AC quasinorms with implications for compactness and embeddings in analysis.
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.