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Sharp pointwise and
uniform estimates for $\bar\partial$
Robert Xin Dong, Song-Ying Li and John N. Treuer |
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Vol. 16 (2023), No. 2, 407–431
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Abstract |
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We use weighted -methods to obtain sharp pointwise estimates for the canonical solution to the equation on smoothly bounded strictly convex domains and the Cartan classical domains when is bounded in the Bergman metric . We provide examples to show our pointwise estimates are sharp. In particular, we show that on the Cartan classical domains of rank the maximum blow-up order is greater than , which was obtained for the unit ball case by Berndtsson. For example, for of type with , the maximum blow-up order is because of the contribution of the Bergman kernel. Additionally, we obtain uniform estimates for the canonical solutions on the polydiscs, strictly pseudoconvex domains and the Cartan classical domains under stronger conditions on . |
Keywords
$L^2$ minimal solution, canonical solution, $\bar\partial$
equation, Cartan classical domain, Bergman kernel, Bergman
metric
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Mathematical Subject Classification
Primary: 32A25
Secondary: 32A36, 32M15, 32W05
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Milestones
Received: 26 May 2020
Revised: 29 March 2021
Accepted: 10 June 2021
Published: 3 May 2023
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Authors |
| Robert Xin Dong | |
| Department of Mathematics University of Connecticut Storrs, CT United States |
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| Song-Ying Li | |
| Department of Mathematics University of California Irvine, CA United States |
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| John N. Treuer | |
| Department of Mathematics University of California Irvine, CA United States |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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