Dual spaces to Orlicz - Lorentz spaces

Abstract

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

arXiv comment25 pages