Dual spaces to Orlicz–Lorentz spaces
Volume 222 / 2014
Abstract
For an Orlicz function $\varphi $ and a decreasing weight $w$, two intrinsic exact descriptions are presented for the norm in the Köthe dual of the Orlicz–Lorentz function space $\varLambda _{\varphi ,w}$ or the sequence space $\lambda _{\varphi ,w}$, equipped with either the Luxemburg or Amemiya norms. The first description is via the modular $\inf\{\int \varphi _*(f^*/|g|)|g|: g\prec w\}$, where $f^*$ is the decreasing rearrangement of $f$, $\prec $ denotes submajorization, and $\varphi _*$ is the complementary function to $\varphi $. The second description is in terms of the modular $\int _I \varphi _*((f^*)^0/w)w$, where $(f^*)^0$ is Halperin's level function of $f^*$ with respect to $w$. That these two descriptions are equivalent results from the identity $\inf\{\int \psi (f^*/|g|)|g|: g\prec w\}=\int _I \psi ((f^*)^0/w)w$, valid for any measurable function $f$ and any Orlicz function $\psi $. An analogous identity and dual representations are also presented for sequence spaces.
