Haj\lasz-Sobolev Imbedding and Extension

Abstract

The author establishes some geometric criteria for a Haj\lasz-Sobolev $\dot M^{s,\,p}_\ball$-extension (resp. $\dot M^{s,\,p}_\ball$-imbedding) domain of ${\mathbb R}^n$ with $n\ge2$, $s\in(0,\,1]$ and $p\in[n/s,\,\infty]$ (resp. $p\in(n/s,\,\infty]$). In particular, the author proves that a bounded finitely connected planar domain $\boz$ is a weak $\alpha$-cigar domain with $\alpha\in(0,\,1)$ if and only if $\dot F^s_{p,\,\infty}({\mathbb R}^2)|_\boz=\dot M^{s,\,p}_\ball(\boz)$ for some/all $s\in[\alpha,\,1)$ and $p=(2-\az)/(s-\alpha)$, where $\dot F^s_{p,\,\infty}({\mathbb R}^2)|_\boz$ denotes the restriction of the Triebel-Lizorkin space $\dot F^s_{p,\,\infty}({\mathbb R}^2) $ on $\boz$.