Optimality of the rearrangement inequality with applications to Lorentz-type sequence spaces

Albiac, Fernando; Ansorena, José L.; Leung, Denny H.; Wallis, Ben

doi:10.7153/mia-2018-21-10

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A dichotomy for subsymmetric basic sequences with applications to Garling spaces

Albiac¹,

Ansorena²,

Dilworth³

et al. 2021

Trans. Amer. Math. Soc.

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Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positionings and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that these spaces have a continuum of subsymmetric basic sequences.

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A dichotomy for subsymmetric basic sequences with applications to Garling spaces

Albiac¹,

Ansorena²,

Dilworth³

et al. 2021

Trans. Amer. Math. Soc.

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Subrearrangement-invariant function spaces

Wallis¹

2019

Preprint

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Rearrangement-invariance in function spaces can be viewed as a kind of generalization of 1-symmetry for Schauder bases. We define subrearrangement-invariance in function spaces as an analogous generalization of 1-subsymmetry. It is then shown that every rearrangement-invariant function space is also subrearrangement-invariant. Examples are given to demonstrate that not every function space on (0, ∞) admits an equivalent subrearrangement-invariant norm, and that not every subrearrangementinvariant function space on (0, ∞) admits an equivalent rearrangementinvariant norm. The latter involves constructing a family of function spaces inspired by D.J.H. Garling, and we further study them by showing that they contain copies of ℓ p .

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Optimality of the rearrangement inequality with applications to Lorentz-type sequence spaces

Abstract: Abstract. We characterize the sequences (w i ) ∞ i=1 of non-negative numbers for whichruns over all non-increasing sequences of non-negative numbers. As a by-product of our work we settle a problem raised in [1] and prove that Garling sequences spaces have no symmetric basis.

Cited by 2 publications

References 2 publications

A dichotomy for subsymmetric basic sequences with applications to Garling spaces

A dichotomy for subsymmetric basic sequences with applications to Garling spaces

Subrearrangement-invariant function spaces

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