Local monotonicity structure of symmetric spaces with applications

Ciesielski, Maciej; Kolwicz, Paweł; Panfil, Agata

doi:10.1016/j.jmaa.2013.07.028

Cited by 27 publications

(53 citation statements)

References 18 publications

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“…Immediately, by definition of K -order continuity and by the above lemma as well as by Lemma 2.5 in [8] in case when E is order continuous we conclude the following corollary.…”

Section: Proof (I) ⇒ (Ii)supporting

confidence: 53%

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Strict K-monotonicity and K-order continuity in symmetric spaces

Ciesielski

2017

Positivity

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This paper is devoted to strict K -monotonicity and K -order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K -monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K -order continuity in a symmetric space E on [0, ∞) implies that the embedding E → L 1 [0, ∞) does not hold. We present a complete characterization of an equivalent condition to K -order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K -monotonicity and K -order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec-Klee property for global convergence in measure in a symmetric space E.

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“…Immediately, by definition of K -order continuity and by the above lemma as well as by Lemma 2.5 in [8] in case when E is order continuous we conclude the following corollary.…”

Section: Proof (I) ⇒ (Ii)supporting

confidence: 53%

“…by Lemma 2.5 in [8] and Lemma 1 for any > 0 there is t > 0 such that for all t ≥ t and n ∈ N we have max{x…”

Section: Proof (Ii) ⇒ (I)mentioning

confidence: 90%

Strict K-monotonicity and K-order continuity in symmetric spaces

Ciesielski

2017

Positivity

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“…Then there exists a measure preserving transformation σ : C → (0, m(C)) such that x * • σ = |x| m-a.e. on C (see Lemma 2.2 in [7]). Passing to a subsequence if necessary, we may assume that a n ≤ m (C) for each n. Take a sequence (A n ) in C such that m (A n ) = a n and x * χ (0,a n ) • σ = |x|χ A n .…”

Section: Lemma 47 Suppose E Is a Quasi-banach Function Space And X ∈mentioning

confidence: 99%

“…However, the studies of global properties are not always sufficient. The geometric structure of a separated point of a Banach space (Banach lattice) has been intensively investigated recently (see [7,8,14,15,19,24,25,27,28]). On the other hand, a symmetrization E ( * ) of a quasi-Banach function space E is an important construction covering several classical classes of Banach function spaces (see [12,13,21,22,[24][25][26] We study Kadec-Klee properties with respect to global (local) convergence in measure.…”

Section: Introductionmentioning

confidence: 99%

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Kadec–Klee properties of some quasi-Banach function spaces

Kolwicz

2018

Positivity

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Best approximation properties in spaces of measurable functions

Ciesielski

Lewicki

2017

Mathematische Nachrichten

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Abstract. We research proximinality of µ-sequentially compact sets and µ-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and µ-compactness of the sets in Banach function spaces. Also the property S is investigated in Fréchet spaces and employed to provide the Kadec-Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fréchet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one-dimensional subspace in sequence Lorentz spaces d(w, 1). Mathematics Subjects Classification: 41A65, 46E30, 46A40.

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Local monotonicity structure of symmetric spaces with applications

Cited by 27 publications

References 18 publications

Strict K-monotonicity and K-order continuity in symmetric spaces

Strict K-monotonicity and K-order continuity in symmetric spaces

Kadec–Klee properties of some quasi-Banach function spaces

Best approximation properties in spaces of measurable functions

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