Let $n\ge2$ and $\phi : [0,\fz) \to [0,\infty)$ be a Young's function satisfying $\sup_{ x>0} \int_0^1\frac{ \phi( t x)}{ \phi(x)}\frac{ dt}{ t^{ n+1} }<\infty. $ We show that Ahlfors $n$-regular domains are Besov-Orlicz ${ \dot { \bf B}}^{ \phi}$ extension domains, which is necessary to guarantee the nontrivially of ${ \dot { \bf B}}^{ \phi}$. On the other hand, assume that $\phi$ grows sub-exponentially at $\fz$ additionally. If $\Omega$ is a Besov-Orlicz ${ \dot { \bf B}}^{ \phi}$ extension domain, then it must be Ahlfors $n$-regular.
Categories: math.FA