Dual spaces to Orlicz-Lorentz spaces

Anna Kamińska ; Karol Leśnik ; Yves Raynaud

Studia Mathematica, Tome 223 (2014), p. 229-261 / Harvested from The Polish Digital Mathematics Library

Résumé

For an Orlicz function φ and a decreasing weight w, two intrinsic exact descriptions are presented for the norm in the Köthe dual of the Orlicz-Lorentz function space $Λ_{φ,w}$ or the sequence space $λ_{φ,w}$, equipped with either the Luxemburg or Amemiya norms. The first description is via the modular $inf{∫ φ⁎(f*/|g|)|g|: g ≺ w}$, where f* is the decreasing rearrangement of f, ≺ denotes submajorization, and φ⁎ is the complementary function to φ. The second description is in terms of the modular $∫_{I} φ⁎((f*)⁰/w)w$,where (f*)⁰ is Halperin’s level function of f* with respect to w. That these two descriptions are equivalent results from the identity $inf{ ∫ ψ(f*/|g|)|g|: g ≺ w} = ∫_{I} ψ((f*)⁰/w)w$, valid for any measurable function f and any Orlicz function ψ. An analogous identity and dual representations are also presented for sequence spaces.

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Publié le : 2014-01-01

Zbl 1322.46021

EUDML-ID : urn:eudml:doc:286639