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Transactions of the American Mathematical Society

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$ \overline\partial\sb b$-equations on certain unbounded weakly pseudo-convex domains


Author: Hyeonbae Kang
Journal: Trans. Amer. Math. Soc. 315 (1989), 389-413
MSC: Primary 35N15; Secondary 32F20
DOI: https://doi.org/10.1090/S0002-9947-1989-0989577-5
MathSciNet review: 989577
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Abstract: We found an explicit closed formula for the relative fundamental solution of $ {\bar \partial _b}$ on the surface $ {H_k} = \{ ({z_1},{z_2}) \in {\mathbb{C}^2}:\operatorname{Im} {z_2} = \vert{z_1}{\vert^{2k}}\} $ . We then make estimates of the relative fundamental solution in terms of the nonisotropic metric associated with the surface. The estimates lead us to the regularity results. We also study the problem of finding weights $ \omega $ so that $ {\bar \partial _b}$ as an operator from $ L_\omega ^2$ to $ {L^2}$ has a closed range. We find the best possible weight among radial weights.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0989577-5
Keywords: Tangential Cauchy Riemann equation, Szegö kernel, relative fundamental solution, nonisotropic metric
Article copyright: © Copyright 1989 American Mathematical Society

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