Orlicz spaces associated to a quasi-Banach function space. Applications to vector measures and interpolation

TL;DR

This paper explores the properties of Orlicz spaces linked to quasi-Banach function spaces derived from vector measures, establishing key compactness characterizations and a de la Vallée-Poussin theorem in this setting.

Contribution

It extends classical compactness and integrability concepts to Orlicz spaces associated with vector measures, providing new tools for analysis in this generalized context.

Findings

Characterization of relatively compact subsets in $L^1(\| m \\|)$

Establishment of a de la Vallée-Poussin type theorem for these spaces

Identification of compact subsets within smaller Orlicz spaces

Abstract

We characterize the relatively compact subsets of $L^1\left(\| m \| \right),$ the quasi-Banach function space associated to the semivariation of a given vector measure $m$ showing that the strong connection between compactness, uniform absolute continuity, uniform integrability, almost order boundedness and L-weak compactness that appears in the classic setting of Lebesgue spaces remains almost invariant in this new context of the Choquet integration. Also we present a de la Vall\'ee-Poussin type theorem in the context of these spaces $L^1\left(\|m\|\right)$ that allows us to locate each compact subset of $L^1\left(\|m\|\right)$ as a compact subset of a smaller quasi-Banach Orlicz space $L^\Phi\left(\|m\|\right)$ associated to the semivariation of the measure $m.$