Orlicz-Besov extension and Ahlfors $n$-regular domains

Abstract

Let $n\ge 2$ and $ϕ: [0,\fz) \to [0,\infty)$ be a Young's function satisfying ${\sup }_{x>0}{\int }_0^1\frac{\varphi (tx)}{\varphi (x)}\frac{dt}{t^{n+1}}<\infty .$ We show that Ahlfors $n$-regular domains are Besov-Orlicz ${\dot{\mathbf{B}}}^{\varphi }$ extension domains, which is necessary to guarantee the nontrivially of ${\dot{\mathbf{B}}}^{\varphi }$. On the other hand, assume that $\varphi$ grows sub-exponentially at $\fz$ additionally. If $\Omega$ is a Besov-Orlicz ${\dot{\mathbf{B}}}^{\varphi }$ extension domain, then it must be Ahlfors $n$-regular.