Dual spaces to Orlicz-Lorentz spaces

Anna Kamińska ; Karol Leśnik ; Yves Raynaud

Studia Mathematica, Tome 222 (2014) no. 3, p. 229.

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 i n f ∫ 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 i n f ∫ 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.

Détail

Classification : 46B10, 46E30, 42B25
Mots-clés : Orlicz-Lorentz spaces, Lorentz spaces, dual spaces, level function, Calderón-Lozanovskii spaces, r.i. spaces