Haj\lasz-Sobolev Imbedding and Extension

Details Haj\lasz-Sobolev Imbedding and Extension Haj\lasz-Sobolev Imbedding and Extension

2010-04-29

arxiv.org/abs/1004.5307v2

J. Math. Anal. Appl. 382 (2011) 577-593

Mathematics

Classical Analysis and ODEs

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Scientific paper

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 $\boz$.

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