Positively Norming Sets in Banach Function Spaces

Abstract. We study compactness and related topological properties in the space L 1 (m) of a Banach space valued measure m when the natural topologies associated to the convergence of the vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L 1 (m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration operator is analyzed in relation with the positive Schur property of L 1 (m). The strong weakly compact generation of L 1 (m) is discussed as well.

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Compactness in L¹of a vector measure

Calabuig

Lajara

Rodríguez

et al. 2014

Studia Math.

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$p$-Regularity and Weights for Operators Between $L^p$-Spaces

Pérez

Tradacete

2020

Z. Anal. Anwend.

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We explore the connection between p-regular operators on Banach function spaces and weighted p-estimates. In particular, our results focus on the following problem. Given finite measure spaces µ and ν, let T be an operator defined from a Banach function space X(ν) and taking values on L p (vdµ) for v in certain family of weights V ⊂ L 1 (µ) + : we analyze the existence of a bounded family of weights W ⊂ L 1 (ν) + such that for every v ∈ V there is w ∈ W in such a way that T : L p (wdν) → L p (vdµ) is continuous uniformly on V . A condition for the existence of such a family is given in terms of p-regularity of the integration map associated to a certain vector measure induced by the operator T .2010 Mathematics Subject Classification. 46E30,46B42,47B10. Key words and phrases. Banach function space; p-regular operator; weighted p-estimate. E. A. Sánchez Pérez gratefully acknowledges support of Spanish Ministerio de Economía, Industria y Competitividad through grant MTM2016-77054-C2-1-P. P. Tradacete gratefully acknowledges support of Spanish Ministerio de Economía, Industria y Competitividad through grants MTM2016-76808-P and MTM2016-75196-P. PreliminariesLet (Ω, Σ, µ) be a finite measure space. Let L 0 (µ) be the space of all measurable real functions on Ω, identifying functions that are equal µ-a.e. We say that a Banach function space over µ is a Banach space X(µ) ⊆ L 0 (µ) that contains all simple functions, and whenever |f | ≤ |g| and g ∈ X(µ), then f ∈ X(µ) with f X(µ) ≤ g X(µ) . For the aim of simplicity, we will write sometimes X instead of X(µ) if the measure does not play a relevant role or is clear in the context. Recall X(µ) is order continuous if for every sequence (f n ) n ∈ X(µ), f n ↓ 0 implies f n X(µ) → 0. In this case, the Köthe dual X(µ) ′ = {g ∈ L 0 (µ) : f g ∈ L 1 (µ) whenever f ∈ X(µ)}, p-REGULAR OPERATORS AND WEIGHTS

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Positively Norming Sets in Banach Function Spaces

Cited by 2 publications

References 26 publications

Compactness in L¹of a vector measure

Compactness in L¹of a vector measure

$p$-Regularity and Weights for Operators Between $L^p$-Spaces

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Positively Norming Sets in Banach Function Spaces

Cited by 2 publications

References 26 publications

Compactness in L1of a vector measure

Compactness in L1of a vector measure

$p$-Regularity and Weights for Operators Between $L^p$-Spaces

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Compactness in L¹of a vector measure

Compactness in L¹of a vector measure