LAUSR

Abstract On Orlicz spaces of measurable functions, the classical Orlicz and Luxemburg norm can be defined by use of the Amemiya formula: ‖ x ‖ Φ o = inf k > 0 ⁡ 1 k ( 1 + I Φ ( k x ) ) and ‖ x ‖ = inf k > 0 ⁡ 1 k max ⁡ { 1 , I Φ ( k x ) } respectively, where Φ is an Orlicz function and I Φ ( x ) = ∫ T Φ ( x ( t ) ) d μ ( 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 = inf k > 0 ⁡ 1 k ( 1 + I Φ p ( k x ) ) 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 = inf k > 0 ⁡ 1 k s ( I Φ ( k x ) ) 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 Holder sense, i.e. u + v ≤ s ( u ) s ⁎ ( v ) for all u , v ≥ 0 . Moreover, it is proved, under some minor assumptions, that the Kothe dual of the subspace E Φ of the Orlicz space equipped with the s-norm ‖ ⋅ ‖ Φ , s is an Orlicz space L Ψ equipped with the s-norm ‖ ⋅ ‖ Ψ , s ⁎ , where the outer function s ⁎ is conjugate to s in the Holder sense and the Orlicz function Ψ is complementary to Φ in the Young sense.