Characterizations of various domains of holomorphy via $\bar{\partial}$ estimates and applications to a problem of Kohn

Abstract

It is shown that the only pseudoconvex sets with smooth boundary in $\mathbf{C}^{n}$ on which $\bar{\partial}$ satisfies Lipschitz smoothing estimates of order $1/2$ are the strongly pseudoconvex ones. Various extensions of this result are made to weakly pseudoconvex domains of finite type and in various norms. It is proved that subelliptic estimates for $\bar{\partial}$ can hold on a pseudoconvex set in $\mathbf{C}^{n}$ only if the domain is of finite type in the sense of Kohn.