L²-Sobolev Theory for ∂ on Domains in \Bbb CPⁿ

2025/07/25 by Shaw, Mei-Chi
#32W05 #35N15 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.19355

Abstract

In this article, we study the range of the Cauchy-Riemann operator ∂ on domains in the complex projective space \BbbCPⁿ. In particular, we show that ∂ does not have closed range in L² for (2,1)-forms on the Hartogs triangle in \BbbCP². We also study the ∂-Cauchy problem on pseudoconvex domains and use it to prove the Sobolev estimates for ∂ on pseudoconcave domains in \BbbCPⁿ.

Citations

• On the L²-Dolbeault cohomology of annuli
• The Diederich-Fornæss exponent and non-existence of Stein domains with Levi-flat boundaries
• The L²-cohomology of a bounded smooth Stein Domain is not necessarily Hausdorff
• On the Hausdorff property of some Dolbeault cohomology groups
• L² Serre Duality on Domains in Complex Manifolds and Applications

Related

• ∂ Sobolev-type inequality and an improved L²-estimate of ∂ on bounded strictly pseudoconvex domains
• Estimates for the ∂-Neumann problem and nonexistence of Levi-flat hypersurfaces in CPⁿ
• Aspects of the L²-Sobolev theory of the ∂-Neumann problem
• Closed range estimates for ∂_b on CR manifolds of hypersurface type
• Weighted Sobolev L^p estimates for homotopy operators on strictly pseudoconvex domains with C² boundary