Orlicz-Besov imbedding and globally $n$-regular domains

Abstract

Denote by $ {\bf\dot B}^{\alpha,\phi}(\Omega)$ the Orlicz-Besov space, where $\alpha\in\mathbb{R}$, $\phi$ is a Young function and $\Omega\subset\mathbb{R}^n$ is a domain. For $\alpha\in(-n,0)$ and optimal $\phi$, in this paper we characterize domains supporting the imbedding ${\bf\dot B}^{\alpha,\phi}(\Omega)$ into $ L^{n/|\alpha|}(\Omega)$ via globally $n$-regular domains. This extends the known characterizations for domains supporting the Besov imbedding ${\bf\dot B} ^s_{pp}(\Omega)$ into $ L^{np/(n-sp)}(\Omega)$ with $s\in(0,1)$ and $1\le p<n/s$. The proof of the imbedding ${\bf\dot B}^{\alpha,\phi}(\Omega)\to L^{n/|\alpha|}(\Omega)$ in globally $n$-regular domains $\Omega$ relies on a geometric inequality involving $\phi$ and $\Omega$ , which extends a known geometric inequality of Caffarelli et al.