The $\bar\partial$-Cauchy Problem on Weakly $q$-Convex Domains in $\Bbb{C}P^n$

Sayed Saber

Let $D$ be a weakly $q$-convex domain in the complex projective space $\Bbb{C}P^n$. In this paper, the (weighted) $\bar\partial$-Cauchy problem with support conditions in $D$ is studied. Specifically, the modified weight function method is used to study the $L^2$ existence theorem for the $\bar\partial$-Neumann problem on $D$. The solutions are used to study function theory on weakly $q$-convex domains via the $\bar\partial$-Cauchy problem.