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

## TL;DR

This paper characterizes Ahlfors n-regular domains as Besov-Orlicz extension domains under certain conditions on the Young's function, linking geometric regularity with functional extension properties.

## Contribution

It establishes a characterization of Ahlfors n-regular domains as Besov-Orlicz extension domains, providing necessary and sufficient conditions based on growth properties of the Young's function.

## Key findings

- Ahlfors n-regular domains are Besov-Orlicz extension domains under certain conditions.
- The necessity of Ahlfors n-regularity for Besov-Orlicz extension domains when the Young's function grows sub-exponentially.
- The paper links geometric regularity with functional extension properties in the context of Besov-Orlicz spaces.

## Abstract

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.