Uniform convexity, reflexivity, supereflexivity and $B$ convexity of generalized Sobolev spaces $W^{1,\Phi }$

Abstract

We investigate Sobolev spaces $W^{1,\Phi }$ associated to Musielak-Orlicz spaces $L^{\Phi }$. We first present conditions for the boundedness of the Voltera operator in $L^{\Phi }$. Employing this, we provide necessary and sufficient conditions for $W^{1,\Phi }$ to contain isomorphic subspaces to ${\ell }^{\infty }$ or ${\ell }^1$. Further we give necessary and sufficient conditions in terms of the function $\Phi$ or its complementary function ${\Phi }^{\ast }$ for reflexivity, uniform convexity, $B$-convexity and superreflexivity of $W^{1,\Phi }$. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces $W^{1,\phi }$ where $\phi$ is an Orlicz function, the variable exponent Sobolev spaces $W^{1,p(\cdot )}$ and the Sobolev spaces associated to double phase functionals.

28 pages