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Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces. (English) Zbl 0906.46026
Let \(X= \{X_1,\dots, X_n\}\) be a system of locally Lipschitz real-valued vector fields in \(\mathbb{R}^n\), \(d(x, y)\) be the Carnot-Carathéodory distance associated to \(X\), \(B(x_0, R)= \{x\in\mathbb{R}^n\mid d(x, x_0)< R\}\). The authors introduce the following basic assumptions: the openness of the balls \(B(x_0, R)\) in the Euclidean topology, the so-called “doubling condition”, and some Poincaré type inequality. Under these not very restrictive conditions some extension properties of Sobolev spaces are investigated and approximation of Sobolev functions by functions which are smooth up to the boundary of a domain is obtained. In particular, for the so-called \((\varepsilon, \delta)\)-domain \(\Omega\subset\mathbb{R}^n\) with the radius of \(\Omega>0\) the following extension result is established.
Let \(1\leq p<\infty\), let \({\mathcal L}^{1,p}(\Omega)= \{f\in L^p(\Omega)\mid X_jf\in L^p(\Omega), j= 1,\dots,m\}\) be the weak Sobolev space endowed with the norm \(\| f\|_{{\mathcal L}^{1,p}(\Omega)}= \| f\|_{L^p(\Omega)}+ \sum^m_{j= 1}\| X_jf\|_{L^p(\Omega)}\). Then there exists a linear operator \({\mathcal E}: {\mathcal L}^{1,p}(\Omega)\to {\mathcal L}^{1,p}(\mathbb{R}^n)\) such that for some \(C>0\) one has for \(f\in {\mathcal L}^{1,p}(\Omega)\),
(i) \({\mathcal E}f(x)= f(x)\) for a.e. \(x\in\Omega\),
(ii) \(\|{\mathcal E}f\|_{{\mathcal L}^{1,p}(\mathbb{R}^n)}\leq C\| f\|_{{\mathcal L}^{1,p}(\Omega)}\).

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35S15 Boundary value problems for PDEs with pseudodifferential operators
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