Orlicz spaces equipped with s-norms

Abstract

On Orlicz spaces of measurable functions, the classical Orlicz and Luxemburg norm can be defined by use of the Amemiya formula: ‖x‖Φo=infk>0⁡1k(1+IΦ(kx)) and ‖x‖=infk>0⁡1kmax⁡{1,IΦ(kx)} respectively, where Φ is an Orlicz function and IΦ(x)=∫TΦ(x(t))dμ(t). Based on this observation, in the last few years a number of papers have been published that dealt with the geometrical properties of Orlicz spaces equipped with the so-called p-Amemiya norms defined by ‖x‖Φ,p=infk>0⁡1k(1+IΦp(kx))1/p, where 1≤p≤∞. The aim of this paper is to present a general and universal method of introducing norms in Orlicz spaces that will cover all the cases mentioned above. Namely, using the concept of outer function, s-norms ‖x‖Φ,s=infk>0⁡1ks(IΦ(kx)) are introduced. It is proved that to each outer function s we can associate an outer function s⁎ that is conjugate to s in the Hölder sense, i.e. u+v≤s(u)s⁎(v) for all u,v≥0. Moreover, it is proved, under some minor assumptions, that the Köthe dual of the subspace EΦ of the Orlicz space equipped with the s-norm ‖⋅‖Φ,s is an Orlicz space LΨ equipped with the s-norm ‖⋅‖Ψ,s⁎, where the outer function s⁎ is conjugate to s in the Hölder sense and the Orlicz function Ψ is complementary to Φ in the Young sense.