Extension in generalized Orlicz--Sobolev spaces

Abstract

We study the existence of an extension operator $\Lambda :W^{1,\phi }(\Omega )\to W^{1,\psi }({\mathbb{R}}^n)$. We assume that $\phi \in {\Phi }_{\mathrm{w}}(\Omega )$ has generalized Orlicz growth, $\psi \in {\Phi }_{\mathrm{w}}({\mathbb{R}}^n)$ is an extension of $\phi$, and that $\Omega \subset {\mathbb{R}}^n$ is an $(\epsilon ,\delta )$-domain. Special cases include the classical constant exponent case, the Orlicz case, the variable exponent case, and the double phase case.