Extension in generalized Orlicz--Sobolev spaces

TL;DR

This paper investigates the conditions under which extension operators exist between generalized Orlicz--Sobolev spaces, covering various special cases like classical, Orlicz, variable exponent, and double phase spaces.

Contribution

It establishes the existence of extension operators in generalized Orlicz--Sobolev spaces under broad conditions, extending previous results to more general growth functions.

Findings

Extension operators exist for generalized Orlicz--Sobolev spaces on $( ext{epsilon, delta})$-domains.

Includes special cases such as classical, Orlicz, variable exponent, and double phase spaces.

Provides a unified framework for extension problems in various generalized Sobolev spaces.

Abstract

We study the existence of an extension operator $\Lambda \colon W^{1,\varphi}(\Omega)\to W^{1,\psi}(\mathbb{R}^n)$. We assume that $\varphi \in \Phi_\mathrm{w}(\Omega)$ has generalized Orlicz growth, $\psi \in \Phi_\mathrm{w}(\mathbb{R}^n)$ is an extension of $\varphi$, 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.