A version of a theorem of Dahlberg for the subelliptic Dirichlet problem
According to B. Dahlberg for each bounded Lipschitz domain contained in \(\mathbb{R}^n\) harmonic and surface measures are mutually absolutely continuous. The authors of the paper under review announce a similar result for the sub-Laplace operator \(L=\sum^m_{j=1}X_j^*X_j\), where the smooth real vector fields \(X=\{X_j\}^m_{j=1}\) satisfy Hörmander's hypoellipticity condition. Instead of Lipschitz domains here a class is defined of domains \(\Omega\) admissible for the Dirichlet problem \((A{\mathcal D}P_X)\) for \(L\), \(\Omega\) being connected, smooth, open bounded and satisfying the uniform outer \(L\)-ball condition. As an application of their main results the authors investigate the Dirichlet problem \(Lu=0\), \(u|_{\partial\Omega}=\varphi\in L_p\), \(1<p<\infty\), where \(\Omega\) is an \(A{\mathcal D}P_X\) domain. They prove the existence of a unique solution \(u\). Moreover, \(u(Q)\) converges nontangentially to \(\varphi(Q)\) for a.e. \(Q\in\partial\Omega\) with respect to an appropriate measure \(d\mu\) on the boundary.
scientific article; zbMATH DE number 1321867