$L^2$ estimates and existence theorems for $\protect \overline{\partial }_b$ on Lipschitz boundaries of $Q$-pseudoconvex domains

Abstract

On a bounded $q$-pseudoconvex domain $\Omega $ in $\mathbb{C}^{n}$ with Lipschitz boundary $b\Omega $, we prove the $L^2$ existence theorems of the $\overline{\partial }_b$-operator on $b\Omega $. This yields the closed range property of $\overline{\partial }_b$ and its adjoint $\overline{\partial }_b^*$. As an application, we establish the $L^2$-existence theorems and regularity theorems for the $\overline{\partial }_b$-Neumann operator.