Dual spaces to Orlicz-Lorentz spaces

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 Λφ;w or the sequence space Λφ;w , equipped with either the Luxemburg or Amemiya norms. The first description is via the modular inf {∫φ∗(f∗/I g I)I g I : g \u3c w }, where f∗ is the decreasing rearrangement of f , \u3c- denotes submajorization, and φ∗is the complementary function to φ. The second description is in terms of the modular ∫Iφ ((f∗)0 /w )w , where (f∗)0 is Halperin\u27s level function of f∗ with respect to w. That these two descriptions are equivalent results from the identity inf {∫ ψ (f∗/I g I)I g I : g \u3c- 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