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Compactness in quasi-Banach function spaces with applications to \(L^1\) of the semivariation of a vector measure

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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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Acknowledgements

We want to thank the anonymous referee for his comments and questions that have contributed to improving the content of this paper and also for pointing out to us the references [12, 15].

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Authors and Affiliations

  1. Dpto. Matemática Aplicada I, Universidad de Sevilla, EUITA, Ctra. de Utrera Km. 1, 41013, Sevilla, Spain

    Ricardo del Campo

  2. Dpto. Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Camino de los Descubrimientos, s/n, 41092, Sevilla, Spain

    Antonio Fernández, Fernando Mayoral & Francisco Naranjo

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  1. Ricardo del Campo
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  2. Antonio Fernández
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  3. Fernando Mayoral
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  4. Francisco Naranjo
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Correspondence to Antonio Fernández.

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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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