Estimates for the ∂̄-neumann Problem and Nonexistence of C Levi-flat Hypersurfaces in Cp*

Let Ω be a pseudoconvex domain with C2 boundary in CPn, n ≥ 2. We prove that the ∂̄-Neumann operator N exists for square-integrable forms on Ω. Furthermore, there exists a number 0 > 0 such that the operators N , ∂̄∗N , ∂̄N and the Bergman projection are regular in the Sobolev space W (Ω) for < 0. The ∂̄-Neumann operator is used to construct ∂̄-closed extension on Ω for forms on the boundary bΩ. This gives solvability for the tangential Cauchy-Riemann operators on the boundary. Using these results, we show that there exist no non-zero L2-holomorphic (p, 0)forms on any domain with C2 pseudoconcave boundary in CPn with p > 0 and n ≥ 2. As a consequence, we prove the nonexistence of C2 Levi-flat hypersurfaces in CPn.