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 an Orlicz-Lorentz function space $Λ_{φ,w}$ or a sequence space $λ_{φ,w}$, equipped with either Luxemburg or Amemiya norms. The first description of the dual norm is given via the modular $\inf\{\intφ_*(f^*/|g|)|g|: g\prec w\}$, where $f^*$ is the decreasing rearrangement of $f$, $g\prec w$ denotes the submajorization of $g$ by $w$ and $φ_*$ is the complementary function to $φ$. The second one is stated in terms of the modular $\int_I φ_*((f^*)^0/w)w$, where $(f^*)^0$ is Halperin's level function of $f^*$ with respect to $w$. That these two descriptions are equivalent results from the identity $\inf\{\intψ(f^*/|g|)|g|: g\prec w\}=\int_I ψ((f^*)^0/w)w$ valid for any measurable function $f$ and Orlicz function $ψ$. Analogous identity and dual representations are also presented for sequence spaces.

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