Dual spaces to Orlicz–Lorentz spaces

Anna Kamińska ¹   ; Karol Leśnik ²   ; Yves Raynaud ³

¹ Department of Mathematics University of Memphis Memphis, TN 38152, U.S.A.
² Institute of Mathematics Faculty of Electrical Engineering Poznań University of Technology Piotrowo 3a 60-965 Poznań, Poland
³ Institut de Mathématiques de Jussieu Université Paris 06-UPMC and CNRS 4 place Jussieu F-75252 Paris Cedex 05, France

Studia Mathematica, Tome 222 (2014) no. 3, pp. 229-261

Résumé

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.

Détail

DOI : 10.4064/sm222-3-3

Mots-clés : orlicz function varphi decreasing weight intrinsic exact descriptions presented norm dual orlicz lorentz function space varlambda varphi sequence space lambda varphi equipped either luxemburg amemiya norms first description via modular inf int varphi * * prec where * decreasing rearrangement nbsp prec denotes submajorization varphi * complementary function nbsp varphi second description terms modular int varphi * * where * halperins level function * respect nbsp these descriptions equivalent results identity inf int psi * prec int psi * valid measurable function orlicz function nbsp psi analogous identity dual representations presented sequence spaces