The extension problem for certain function spaces involving fractional orders of differentiability
Let n be a positive integer, and let \(\Omega\subseteq R^ n\) be an open connected domain. If \(1<p<\infty\), and k is a positive integer, then the Sobolev space \(L^ p_ k(\Omega)\) is defined by \(L^ p_ k(\Omega)=\{f\in L^ 1_{loc}(\Omega):D^{\beta}(f)\in L^ p(\Omega)\quad for\quad| \beta| \leq k\},\) and \(\| f\|_{L^ p(\Omega)}\) is defined to be \(\sum_{0<|\beta |\leq k}\| D^{\beta}(f)\|_{L^ p(\Omega)}\). For \(\epsilon >0\), \(\delta >0\), \(\Omega\) is said to be a (\(\epsilon\),\(\delta)\)-domain if, for x, y in \(\Omega\) with \(| x-y| <\delta\), there is a rectifiable path \(\gamma\) such that (i) \(l(\gamma)\leq| x-y| /c\); (ii) d(z,\(\partial\Omega )\geq\epsilon \) in\(f(| z-x|,| z- y|)\); \(z\in\gamma \). If \(N_{\alpha}(f)(x)=\inf_{P}\sup_{x\in Q\subseteq\Omega }l(Q)^{-n-\alpha}\int_{Q}| f(y)-P(y)| dy,\) where P varies over polynomials of degree \(\leq m\), then \({\mathfrak N}^ p_{\alpha}(\Omega)=\{f\in L^ p(\Omega):N_{\alpha}(f)\in L^ p(\Omega)\};\quad\| f\|_{{\mathfrak N}^ p_{\alpha}(\Omega)}=\| f\|_{L^ p(\Omega)}+\| N_{\alpha}(f)\|_{L^ p(\Omega)}.\) In the main result of this paper, the author shoes that if \(\Omega\subseteq R^ n\) is an open, connected (\(\epsilon\),\(\delta)\)- domain, then \(\Omega\) is an extension domain for \({\mathfrak N}^ p_{\alpha}\) for all p, \(1<p<\infty\), and \(\alpha >0\), so that there is an extension operator \(\Lambda_ N\) such that, for \(0<\alpha <N\), \(\|\Lambda_ N(f)\|_{{\mathfrak N}^ p_{\alpha}(\Omega)}\leq C_{p,\alpha}\| f\|_{{\mathfrak N}^ p_{\alpha}(\Omega)}\). Earlier special cases of the main theorem include results of \textit{A. P. Calderón}, Proc. Symp. Pure Math. 4, 33-49 (1961; Zbl 0195.411), \textit{E. M. Stein}, ''Singular integrals and differentiability properties of functions'', (1970; Zbl 0207.135); and \textit{P. W. Jones}, Acta Math. 147, 71-88 (1981; Zbl 0489.30017).
scientific article; zbMATH DE number 3873990
scientific article