$L^2$-Sobolev Theory for $\bar{\partial }$ on Domains in ${\mathbb{CP}}^n$

Abstract

In this article, we study the range of the Cauchy-Riemann operator $\bar{\partial }$ on domains in the complex projective space ${\mathbb{CP}}^n$. In particular, we show that $\bar{\partial }$ does not have closed range in $L^2$ for (2,1)-forms on the Hartogs triangle in ${\mathbb{CP}}^2$. We also study the $\bar{\partial }$-Cauchy problem on pseudoconvex domains and use it to prove the Sobolev estimates for $\bar{\partial }$ on pseudoconcave domains in ${\mathbb{CP}}^n$.

accepted for publication in the Special Issue of Journal of Geometric Analysis in honor of Joe Kohn. Typo corrected