Uniform Rotundity And Locally Uniform Rotundity Of Orlicz-Lorentz Spaces

Since A.Kaminska gave the notion of Orlicz-Lorentz space in 1990,many conclu-sions about the Orlicz-Lorentz space with the Luxemburg norm have been obtained. Wu and Ren's giving the Orlicz norm of Orlicz-Lorentz space in 1999,the results about the Orlicz-Lorentz space with this norm was sporadic,and lack of system. The main work of this paper is:1.Gave the characterization of uniform rotundity in Orlicz-Lorentz sequence space with the Orlicz norm.2.Gave the characterization of locally uniform rotundity in Orlicz-Lorentz function space with the Orlicz norm.We verified that:Orlicz-Lorentz sequence space with the Orlicz norm is uniform rotundity iff(3)φis uniform rotundity on [0,q(ψ-1(1/(ω(1))))]. Letωbe regular for large arguments.ThenΛ(?).ω[0,∞)is locally uniform rotundity (LUR) if and only if the following two conditions are satisfied:(i)φp is strictly convex on [0,∞)(ii)φ∈Δ2 andψ∈A2.This paper has three chapters, each of them investigated different aspect respec-tively.The first chapter mainly states the fundamental theories of Orlicz-Lorentz spaces, respectively some important basic theorems of Orlicz-Lorentz sequence space.The second chapter keeps on considering the sequence spaces.Reference to the theory of uniform rotundity in function space with the Luxemburg norm,we give the characterization of uniform rotundity in sequence spaces with the Orlicz norm.The work of this chapter is difficult and profound.In the last chapter, we turn to the function spaces.It is also a difficult chapter. We mainly research the characterization of locally uniform rotundity in Orlicz-Lorentz spaces with the Orlicz norm.