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

2019/10/29 by del Campo, Ricardo, Fernández, Antonio, Mayoral, Fernando +1
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 46E30 #Secondary 46G10

paper · doi:10.48550/arxiv.1910.13183

Abstract

We characterize the relatively compact subsets of L¹(‖ m ‖ ), 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 Vallée-Poussin type theorem in the context of these spaces L¹(‖m‖) that allows us to locate each compact subset of L¹(‖m‖) as a compact subset of a smaller quasi-Banach Orlicz space L^Φ(‖m‖) associated to the semivariation of the measure m.

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