Orlicz spaces equipped with s-norms
Orlicz spaces equipped with s-norms
复制标题
DOI：
10.1016/j.jmaa.2019.123659
复制
发表时间：
2020-03
影响因子：
1.3
通讯作者：
M. Wisła
中科院分区：
文献类型：
--
作者：
M. Wisła
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 Hölder sense, ie 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 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 Hölder sense and the Orlicz function Ψ is complementary to Φ in the Young sense.
