The Cauchy–Riemann equations on product domains

By Debraj Chakrabarti, Mei-Chi Shaw. 2009

55 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e08aefacb7e52e92394dfc5a1199709303d5b6e3

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In Applied Mathematics

1. A. Naber
2. Jeff Cheeger
3. Liwei Chen
4. G. Tian
5. Andrea Mondino
6. Nicola Gigli
7. Richard Bamler
8. Bing Wang

Citations in the corpus

• Lower bounds on Ricci curvature and quantitative behavior of singular sets (2011, background): “Unlike ours, it utilizes essentially a rigidity result for almost complex structures; see [10].”
• The $L^p$ boundedness of the Bergman projection for a class of bounded Hartogs domains (2013, background): “papers, [9] and [10], Krantz and Peloso show that the Bergman projection for the two dimensional non-smooth worm domain is bounded only when p is in a range depending on the winding of the domain. In [3] and [4], Chakrabarti and Shaw focus on the ∂- equation and the corresponding Sobolev regularitie”