Classification and geometrical properties of the Xθ(⋠)-valued function spaces

This paper is devoted to investigate the Xθ(⋅)-valued function spaces. Based on the notions of bounded topological lattice, Banach space net, continuous modular net and the dual space net, we divide Xθ(⋅)-valued function spaces into two classes: norm-modular spaces and modular-modular spaces. For the first class, we study the separability of L+p(⋅)(I,Xθ(⋅)), give a representation of the dual space L+p(⋅)(I,Xθ(⋅))⁎, find sufficient conditions of the equality Lp(⋅)(I,Xθ(⋅))=L+p(⋅)(I,Xθ(⋅)), and prove the reflexivity of Lp(⋅)(I,Xθ(⋅)) under some reasonable conditions. And for the second class, we prove the completeness and uniform convexity of Lρθ(⋅)(I,Xθ(⋅)) in suitable situations. To show the naturality and rationality of these results, some concrete function spaces such as Lpt±(I,Lp(x,t)(Ω)), Lpt(I,W1,p(x,t)(Ω)) and Lpt(I,W01,p(x,t)(Ω)) together with LPp(t)(I,Lp(x,t)(Ω)) are taken into account.