Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm (Q765232)

Let \(\Phi\) and \(\Psi\) be a Musielak-Orlicz function and its complementary function in the sense of Young, respectively. The main result of this paper concerns weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm (see Theorem 2 and Lemmas 3-6). Namely, it is shown that if the dual space \(X^{\ast}\) of a Banach space \(X\) has the Radon-Nykodym property, then the space \(L_{M}(X)\) is weakly uniformly rotund if and only if \(M\) and \(N\) satisfy the \(\Delta_{2}\) condition, \(M(t,u)\) is strictly convex with respect to \(u\) for \(\mu\)-a.e.\ \(t\in T\) and \(X\) is a weakly uniformly rotund space. On the basis of this result criteria for uniform Gateâux differentiability of Musielak-Orlicz spaces equipped with the Orlicz norm are deduced.