Uniform convexity, reflexivity, superreflexivity and B convexity of generalized Sobolev spaces W1,Φ

Abstract

We investigate Sobolev spaces W1,Φ associated to Musielak-Orlicz spaces LΦ. We first present conditions for the boundedness of the Voltera operator in LΦ. Employing this, we provide necessary and sufficient conditions for W1,Φ to contain isomorphic subspaces to ℓ∞ or ℓ1. Further we give necessary and sufficient conditions in terms of the function Φ or its complementary function Φ⁎ for reflexivity, uniform convexity, B-convexity and superreflexivity of W1,Φ. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces W1,φ where φ is an Orlicz function, the variable exponent Sobolev spaces W1,p(⋅) and the Sobolev spaces associated to double phase functionals.