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

TL;DR

This paper characterizes geometric and reflexive properties of generalized Sobolev spaces $W^{1, ext{ extPhi}}$ linked to Musielak-Orlicz spaces, using conditions on the defining functions and the boundedness of the Voltera operator.

Contribution

It provides necessary and sufficient conditions for reflexivity, uniform convexity, and superreflexivity of these generalized Sobolev spaces based on the properties of the function $ extPhi$.

Findings

Conditions for the boundedness of the Voltera operator in $L^ extPhi$.

Criteria for $W^{1, extPhi}$ to contain isomorphic subspaces to $ ext{ extl}^ extinfty$ or $ extl^ ext1$.

Characterizations of reflexivity, uniform convexity, and superreflexivity in terms of $ extPhi$ and $ extPhi^*$.

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 $\Phi^*$ 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.