Estimates for the $\bar{\mathrm{\partial }}$-Neumann problem and nonexistence of Levi-flat hypersurfaces in $C{P}^n$

Let $\mathrm{\Omega }$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$\mathrm{.}Weprovethatthe$\bar\partial-Neumann operator $N$ exists for $(p,q)$-forms on $\mathrm{\Omega }$. Furthermore, there exists a $t_0>0$ such that the operators $N$, ${\bar{\mathrm{\partial }}}^{\ast }N$, $\bar{\mathrm{\partial }}N$ and the Bergman projection are regular in the Sobolev space $W^t(\bar{\mathrm{\Omega }})$ 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 $\mathbb{C}{P}^n$. We also show that there exist no non-zero $L^2$-holomorphic $(p,0)$-forms on any pseudoconcave domain in $\mathbb{C}{P}^n$ with $p>0$.