This is an announcement for the paper "Dual spaces to Orlicz - Lorentz spaces" by Anna Kaminska, Karol Lesnik, and Yves Raynaud.
Abstract: For an Orlicz function $\varphi$ and a decreasing weight $w$, two intrinsic exact descriptions are presented for the norm in the K"othe dual of an Orlicz-Lorentz function space $\Lambda_{\varphi,w}$ or a sequence space $\lambda_{\varphi,w}$, equipped with either Luxemburg or Amemiya norms. The first description of the dual norm is given via the modular $\inf{\int\varphi_*(f^*/|g|)|g|: g\prec w}$, where $f^*$ is the decreasing rearrangement of $f$, $g\prec w$ denotes the submajorization of $g$ by $w$ and $\varphi_*$ is the complementary function to $\varphi$. The second one is stated 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 Orlicz function $\psi$. Analogous identity and dual representations are also presented for sequence spaces.
Archive classification: math.FA
Mathematics Subject Classification: 42B25, 46B10, 46E30
Remarks: 25 pages
Submitted from: klesnik@vp.pl
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1403.1505
or