The Cauchy-Riemann equations on product domains

2009/10/31 by Debraj Chakrabarti, Chakrabarti, Debraj, Mei-Chi Shaw +1
Mathematics · #32W05 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:32W05

paper · pdf · doi:10.48550/arxiv.0911.0103

Some typos fixed

arxiv created 2010/05/10 · arxiv updated 2010/05/11

Abstract

We establish the L² theory for the Cauchy-Riemann equations on product domains provided that the Cauchy-Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on (p,1)-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.

Related

• Uniform estimates of the Cauchy-Riemann equation on product domains
• Product domains, Multi-Cauchy transforms, and the ∂ equation
• Sobolev regularity of the canonical solutions to ∂ on product domains
• L²-Sobolev Theory for ∂ on Domains in \Bbb CPⁿ
• Uniform estimates for the canonical solution to the ∂-equation on product domains