Local solvability for \(\square_b\) on degenerate CR manifolds and related systems
In an abstract CR-manifold \(M\), consider the Kohn Laplacian \(\square_ b^{(q)}\) acting on \((0,q)\)-forms \[ \square_ b^{(q)}=\bar\partial_ b\bar\partial_ b^*+\bar\partial_ b^*\bar\partial_ b. \] If the Levi form of \(M\) is degenerate and has a kernel of constant dimension, the author shows that for any point \(x_0\in M\) and any \(\delta>0\), there exists an open neighborhood \(\Omega_\delta\) of \(x_0\) such that \[ \|u\|_0\leq\delta(\square_ b^{(0,q)}u,u) \] for all \(u\in\mathcal{D}_{(0,q)}(\Omega_\delta)\). Here, \((\cdot,\cdot)\) denotes the \(L^2\) inner product, \(\|\cdot\|\) the \(L^2\) norm and \(\mathcal{D}_{(0,q)}(\Omega_\delta)\) is the space of smooth \((0,q)\)-forms with compact support in \(\Omega_\delta\). Moreover, it is also shown that the Kohn Laplacian is locally solvable on such \(M\). This result follows from a more general one concerning a class of systems with double characteristics and scalar principal symbol.
scientific article; zbMATH DE number 5566190
scientific article