Extension of differentiable functions beyond the boundary of the domain on Carnot groups
Let \(D\) be a domain of a Carnot group \(\mathbb G\), which is a special Lie group introduced by \textit{P. Pansu} [Ann. Math. (2) 129, 1--60 (1989; Zbl 0678.53042)]. The author investigates the problems of extension of functions in Sobolev spaces \(W_p^1(D)\) and \(L_p^1(D)\). It is shown that the \((\varepsilon,\delta)\)-condition on the domain \(D\) is sufficient for the existence of bounded extension operators from spaces \(W_p^1(D)\) and \(L_p^1(D)\) to spaces \(W_p^1(\mathbb G)\) and \(L_p^1(\mathbb G)\), respectively. The author also obtains some geometric conditions on the domain \(D\) which are necessary for the existence of a bounded extension operator. Furthermore the paper contains some results on the geometry of Carnot groups. For example, the author proves that a ball in the Carnot-Carathéodory metric on a Carnot group is a uniform domain.
scientific article; zbMATH DE number 1047526