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

Abstract

Let $\Omega$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb{C}{P}^n$. We prove that the $\bar{\partial }-Neumannoperator$N$existsfor$(p,q)$-formson$Ω$.Furthermore,thereexistsa$t_0>0$suchthattheoperators$N$,$\bar\partial^*N$,$\bar\partial N$andtheBergmanprojectionareregularintheSobolevspace$W^t (\barΩ) $for$t<t_0$.Theboundaryestimatesabovehaveapplicationsincomplexgeometry.Weusetheestimatestoprovethenonexistenceof$C^{2, α}$realLevi-flathypersurfacesin$\mathbb CP^n$.Wealsoshowthatthereexistnonon-zero$L^2$-holomorphic$(p, 0)$-formsonanypseudoconcavedomainin$\mathbb CP^n$with$p > 0$.