Composition operators in Orlicz-Sobolev spaces
Michał Borowski, Andrea Cianchi
TL;DR
This work analyzes the continuity of the Nemytskii operator $T_f(u)=f(u)$ on Orlicz-Sobolev spaces, including isotropic $W^{1,A}(\Omega)$ and homogeneous $V^{1,A}_0(\Omega)$, as well as anisotropic variants defined by $n$-dimensional Young functions $\Phi$. By leveraging sharp Orlicz-Sobolev embeddings and modular topology, the authors establish modular-continuity of $T_f$ for Lipschitz $f$ on arbitrary open $\Omega$, with $|\,\Omega|$ finite or infinite (the latter requiring $f(0)=0$). They further derive sharp continuity results $T_f:W^{1,A}(\Omega)\to W^{1,B}(\Omega)$ under a growth condition $|f'(t)|\le\kappa E(\kappa|t|)$, where $B$ is tied to $A$, $E$, and the dimension via optimal embeddings; analogous results hold in homogeneous and anisotropic settings, using $\Phi_n$, $\Phi_\circ$, and the linking function $\vartheta$. The paper also demonstrates novel results even for standard anisotropic Sobolev spaces (e.g., orthotropic forms $\Phi(\xi)=\sum_i|\xi_i|^{p_i}$) and discusses modular versus norm topologies in the absence of the $\Delta_2$-condition. Overall, the results extend and sharpen classical Sobolev composition theory to non-standard growth spaces, with implications for nonlinear PDEs and variational problems in non-homogeneous media.
Abstract
The continuity of the Nemytskii operator between Orlicz-Sobolev spaces is investigated. Natural Orlicz-Sobolev versions of classical results for standard Sobolev are established. The results presented not only extend the latter, but also improve them in borderline situations. Anisotropic Orlicz-Sobolev spaces are included in our analysis. The results offered for this class of spaces are new even for customary anisotropic Sobolev spaces.