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

Abstract

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