Abstract
We characterize the relatively compact subsets of the order continuous part \(E_a\) of a quasi-Banach function space E showing that the strong connection between compactness, uniform absolute continuity, uniform integrability, almost order boundedness and L-weak compactness that appears in the classical setting of Lebesgue spaces remains almost invariant in this new context under mild assumptions. We also present a de la Vallée–Poussin type theorem in this context that allows us to locate each compact subset of \(E_a\) as a compact subset of a smaller quasi-Banach Orlicz space \(E^\varPhi .\) Our results generalize the previous known results for the Banach function spaces \(L^1(m)\) and \(L^1_w(m)\) associated to a vector measure m and moreover they can also be applied to the quasi-Banach function space \(L^1\left( \Vert m \Vert \right) \) associated to the semivariation of m.
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This research has been partially supported by La Junta de Andalucía (Spain) under the Grant FQM-133.
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Campo, R.d., Fernández, A., Mayoral, F. et al. Compactness in quasi-Banach function spaces with applications to \(L^1\) of the semivariation of a vector measure. RACSAM 114, 112 (2020). https://doi.org/10.1007/s13398-020-00840-4
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DOI: https://doi.org/10.1007/s13398-020-00840-4
Keywords
- Orlicz spaces
- Vector measure
- Semivariation
- Uniform integrability
- Uniform absolute continuity
- Compactness
- De la Vallée–Poussin’s theorem