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

Mei-Chi Shaw

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

Complex Variables 2025-07-29 v2

Authors: Mei-Chi Shaw

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 $\Bbb{CP}^n$ .

Keywords

cauchy-riemann operator complex projective space hartogs triangle closed range pseudoconvex domain pseudoconcave domain sobolev estimates

Cite

@article{arxiv.2507.19355,
  title  = {$L^2$-Sobolev Theory for $\bar\partial$ on Domains in $\Bbb {CP}^n$},
  author = {Mei-Chi Shaw},
  journal= {arXiv preprint arXiv:2507.19355},
  year   = {2025}
}

Comments

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

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