L\({}^ 2\) existence theorems for the \({\bar \partial}_ b\)-Neumann problem on strongly pseudoconvex CR manifolds
The author studies the \({\bar \partial}_ b\)-Neumann problem along the lines of the \({\bar \partial}\)-Neumann problem developed by Spencer, Morrey, and Kohn for strongly pseudo-convex complex manifolds. The author proves that the \({\bar \partial}_ b\)-Laplacian or Kohn's Laplacian, \(\square_ b={\bar \partial}_ b{\bar \partial}^*_ b+{\bar \partial}^*_ b{\bar \partial}_ b\) completed with a pair of natural boundary conditions on the boundary of an open subset (in a strongly pseudoconvex manifolds in \({\mathbb{C}}^ n)\), in the Hilbert space, has closed range when it acts on (0,q) forms, \(1\leq q\leq n-3;\) then there exists a bounded inverse operator of \(\square_ b\), and the Hodge decomposition theorem for \({\bar \partial}_ b\) holds.
scientific article; zbMATH DE number 4204789
scientific article