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 $n\ge 2$, $s\in (0,1]$ 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 $p=(2-\az)/(s-α)$, where $\dot F^s_{p,\,\infty}({\mathbb R}^2)|_\boz$ denotes the restriction of the Triebel-Lizorkin space ${\dot{F}}_{p,\infty }^s({\mathbb{R}}^2)$ on $\boz$.

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