Estimates for the partial derivative Neumann problem and nonexistence of C-2 Levi-flat hypersurfaces in CPn

摘要

Let Omega be a pseudoconvex domain with C-2 boundary in CPn, n greater than or equal to 2. We prove that the (&PARTIAL;) over bar -Neumann operator N exists for square-integrable forms on Omega. Furthermore, there exists a number epsilon(0) > 0 such that the operators N, (&PARTIAL;) over bar* N, (&PARTIAL;) over barN and the Bergman projection are regular in the Sobolev space W-epsilon(Omega) for epsilon < ε(0). The <(partial derivative)over bar>-Neumann operator is used to construct (&PARTIAL;) over bar -closed extension on Omega for forms on the boundary bOmega. This gives solvability for the tangential Cauchy-Riemann operators on the boundary. Using these results, we show that there exist no non-zero L-2-holomorphic (p, 0)-forms on any domain with C-2 pseudoconcave boundary in CPn with p > 0 and n greater than or equal to 2. As a consequence, we prove the nonexistence of C-2 Levi-flat hypersurfaces in CPn.