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Extendability and the \(\overline{\partial }\) operator on the Hartogs triangle

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In this paper it is shown that the Hartogs triangle \({\mathbf{T}}\) in \({\mathbf{C}}^2\) is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of \({\mathbf{T}}\).

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Almut Burchard

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06290, USA

    Joshua Flynn & Guozhen Lu

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

    Joshua Flynn, Guozhen Lu & Mei-Chi Shaw

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  1. Almut Burchard
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  2. Joshua Flynn
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  3. Guozhen Lu
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  4. Mei-Chi Shaw
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Correspondence to Mei-Chi Shaw.

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A. Burchard is partially supported by an NSERC discovery grant. G. Lu and J. Flynn are partially supported by the Simons foundation. M.-C. Shaw is partially supported by NSF grants. A. Burchard and M.-C. Shaw would like to thank the Banff International Research Station for its kind hospitality during a 2019 workshop which facilitated this collaboration. We would also like to thank Christine Laurent-Thiébaut for her helpful comments.

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Burchard, A., Flynn, J., Lu, G. et al. Extendability and the \(\overline{\partial }\) operator on the Hartogs triangle. Math. Z. 301, 2771–2792 (2022). https://doi.org/10.1007/s00209-022-03008-5

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