Extension in generalized Orlicz--Sobolev spaces

We study the existence of an extension operator $Λ\colon W^{1,φ}(Ω)\to W^{1,ψ}(\mathbb{R}^n)$. We assume that $φ\in Φ_\mathrm{w}(Ω)$ has generalized Orlicz growth, $ψ\in Φ_\mathrm{w}(\mathbb{R}^n)$ is an extension of $φ$, and that $Ω\subset\mathbb{R}^n$ is an $(ε,δ)$-domain. Special cases include the classical constant exponent case, the Orlicz case, the variable exponent case, and the double phase case.

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