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Astashkin, S. V.; Strakhov, S. I.

On disjointly homogeneous Orlicz-Lorentz spaces. (English. Russian original) Zbl 1467.46026

Math. Notes 108, No. 5, 631-642 (2020); translation from Mat. Zametki 108, No. 5, 643-656 (2020).
Summary: A characterization of disjointly homogeneous Orlicz-Lorentz function spaces \(\Lambda_{\varphi,w}\) is obtained. It is used to find necessary and sufficient conditions for an analog of the classical Dunford-Pettis theorem about the equi-integrability of weakly compact sets in \(L_1\) to hold in the space \(\Lambda_{\varphi,w}\). It is also shown that there exists an Orlicz function \(\Phi\) with the upper Matuszewska-Orlicz index equal to 1 for which such an analog in the space \(\Lambda_{\Phi,w}\) does not hold. This answers a recent question of Leśnik, Maligranda, and Tomaszewsk [K. Leśnik et al., “Weakly compact sets and weakly compact pointwise multipliers in Banach function lattices”, Preprint (2019), arXiv:1912.08164].

Cited in 2 Documents

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

Orlicz-Lorentz space; Orlicz space; Orlicz function; symmetric space; disjointly homogeneous space; weakly compact set; Matuszewska-Orlicz indices
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References:

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