The Criterion Of The Fully K-convexity And Locally Uniform Rotundity Of Orlicz-Lorentz Spaces With The Orlicz Norm

A.Kaminska gave the notion of Orlicz-Lorentz spaces in 1990. The spaces not only can be used as a model of the symmetric spaces, but also play an important role in the interpolation theory. In recent years, more and more mathematicians became interested in the spaces. Many conclusions about the Orlicz-Lorentz spaces with the Luxemburg norm have been obtained. Since Wu and Ren's giving the Orlicz norm of Orlicz-Lorentz spaces in 1999, the results about the Orlicz-Lorentz spaces with this norm was sporadic, and lack of system.So we will continue to study the Orlicz-Lorentz spaces with the Orlicz norm.This paper has three chapters:The first chapter mainly states the fundamental theories of Orlicz-Lorentz spaces.The second chapter gives the characterization of locally uniform rotundity in sequence spaces with the Orlicz norm, we obtain the following theorem:Theorem 2.1λφ,ω°is locally uniform rotundity (LUR) if and only if the following two conditions are satisfied:(ⅰ)φis strictly convex on [0,γ], where(ⅱ)φ∈δ2andψ(?)δ2. In the last chapter, We mainly research the characterization of fully K-convex in Orlicz-Lorentz spaces with the Orlicz norm, and give the essential characterization of the fullyκ-convexity in the spaces, and the following theorem is obtained:Theorem 3.1 Orlicz-Lorentz spaces equipped with Orlicz norm A°/ψ,ω(0,∞) (respec-tively. A°/ψ,ω(0,1)) is fully K-convex for all K≥2, if and only if bothψandψsatisfies the A2 condition (respectively, A2 condition for large values),ψis strictly convex, and J∞/0ω(t)dt=∞(respectively,ω(t)>0 for all 0<t<1).