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Sobolev regularity of the \(\overline{\partial }\)-equation on the Hartogs triangle

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Abstract

The regularity of the \(\overline{\partial }\)-problem on the domain \(\{\left|{z_1}\right|\!<\!\left|{z_2}\right|\!<\!1\}\) in \(\mathbb C ^2\) is studied using \(L^2\)-methods. Estimates are obtained for the canonical solution in weighted \(L^2\)-Sobolev spaces with a weight that is singular at the point \((0,0)\). In particular, the singularity of the Bergman projection for the Hartogs triangle is contained at the singular point and it does not propagate.

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Acknowledgments

The authors thank the referee for his detailed comments and suggestions. D. Chakrabarti thanks Dr. S. Gorai for pointing out an error in the first version of this paper, and Prof. M. Vanninathan for helpful hints on Sobolev spaces on nonsmooth domains. He also thanks Prof. M. Ramaswamy, the Dean of the TIFR Centre for Applicable Mathematics, for her active support of this research.

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Authors and Affiliations

  1. TIFR Centre for Applicable Mathematics, Sharada Nagar, Chikkabommasandra, Bengaluru, 560065, India

    Debraj Chakrabarti

  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Mei-Chi Shaw

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  1. Debraj Chakrabarti
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  2. Mei-Chi Shaw
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Correspondence to Debraj Chakrabarti.

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Dedicated to the memory of Prof. Jianguo Cao.

M.-C. Shaw is partially supported by NSF grants.

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Chakrabarti, D., Shaw, MC. Sobolev regularity of the \(\overline{\partial }\)-equation on the Hartogs triangle. Math. Ann. 356, 241–258 (2013). https://doi.org/10.1007/s00208-012-0840-y

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  • DOI: https://doi.org/10.1007/s00208-012-0840-y

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