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

TL;DR

This paper investigates the behavior of the $ar ext{d}$ operator in $L^2$ spaces on complex projective domains, revealing non-closed range phenomena and establishing Sobolev estimates for pseudoconcave domains in $ ext{CP}^n$.

Contribution

It provides new insights into the $L^2$-Sobolev theory for $ar ext{d}$ on domains in $ ext{CP}^n$, including non-closed range results and Sobolev estimates for pseudoconcave domains.

Findings

$ar ext{d}$ does not have closed range in $L^2$ for (2,1)-forms on the Hartogs triangle in $ ext{CP}^2$

Established Sobolev estimates for $ar ext{d}$ on pseudoconcave domains in $ ext{CP}^n$

Analyzed the $ar ext{d}$-Cauchy problem on pseudoconvex domains

Abstract

In this article, we study the range of the Cauchy-Riemann operator $\bar\partial$ on domains in the complex projective space $\Bbb{CP}^n$. In particular, we show that $\bar\partial$ does not have closed range in $L^2$ for (2,1)-forms on the Hartogs triangle in $\Bbb{CP}^2$. 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 $\Bbb{CP}^n$.