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

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 $\Phi^*$ for reflexivity, uniform convexity, $B$-convexity and superreflexivity of $W^{1,\Phi}$. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces $W^{1,\varphi}$ where $\varphi$ is an Orlicz function, the variable exponent Sobolev spaces $W^{1,p(\cdot)}$ and the Sobolev spaces associated to double phase functionals.