Trace theorem in Sobolev spaces associated with Hörmander's vector fields (Q2759770)

Let \(\Omega\subset \mathbb{R}^d\) \((d\geq 2)\) be an open domain, and let \(P= (P_1,\dots, P_n)\) be a smooth real vector field system satisfying the Hörmander condition of order 2. The Sobolev space \(H^k(\Omega, P)\) is defined as the space of functions \(f\in L^2(\Omega)\) such that \(P^Jf\in L^2(\Omega)\) for \(|J|\leq k\). Let \(\Sigma\subset \Omega\) be a smooth surface and let \(\chi\) be the \(C^\infty\)-module of vector fields spanned by \(P\). The space \(H^k(\Sigma, P_\Sigma)\) is considered, where \(P_\Sigma\) is a generator system of \(\chi_\Sigma\). It is proved that the trace operator \(\gamma_\Sigma\) mapping \(H^k(\Omega, P)\) into \([H^k(\Sigma, P_\Sigma)\cap H^{k/2}(\Sigma), L^2(\Sigma)]_{1/(2k)}\) is continuous and surjective. An analogous result is obtained for the trace operator \(\gamma_{\Sigma_0}\) on \(\Sigma_0= \{(x,y,s)\in \mathbb{R}^{2k+1}; s=0\}\) connected with the Heisenberg group \(H^d\).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00056].