Estimates for the $\bar\partial$-Neumann problem and nonexistence of Levi-flat hypersurfaces in $CP^n$

Let $\Omega$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$. We prove that the $\bar\partial-Neumann operator$N$exists for$(p,q)$-forms on$\Omega$. Furthermore, there exists a$t_0>0$such that the operators$N$,$\bar\partial^*N$,$\bar\partial N$and the Bergman projection are regular in the Sobolev space$W^t (\bar{\Omega}) $for$t<t_0$. The boundary estimates above have applications in complex geometry. We use the estimates to prove the nonexistence of$C^{2, \alpha}$real Levi-flat hypersurfaces in$\mathbb CP^n$. We also show that there exist no non-zero$L^2$-holomorphic$(p, 0)$-forms on any pseudoconcave domain in$\mathbb CP^n$with$p > 0$.