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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weighted Bergman spaces and the $ \partial-$equation


Author: Bo-Yong Chen
Journal: Trans. Amer. Math. Soc. 366 (2014), 4127-4150
MSC (2010): Primary 32A25, 32A36, 32A40, 32W05
Published electronically: March 26, 2014
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Abstract: We give a Hörmander type $ L^2-$estimate for the $ \bar {\partial }-$equation with respect to the measure $ \delta _\Omega ^{-\alpha }dV$, $ \alpha <1$, on any bounded pseudoconvex domain with $ C^2-$boundary. Several applications to the function theory of weighted Bergman spaces $ A^2_\alpha (\Omega )$ are given, including a corona type theorem, a Gleason type theorem, together with a density theorem. We investigate in particular the boundary behavior of functions in $ A^2_\alpha (\Omega )$ by proving an analogue of the Levi problem for $ A^2_\alpha (\Omega )$ and giving an optimal Gehring type estimate for functions in $ A^2_\alpha (\Omega )$. A vanishing theorem for $ A^2_1(\Omega )$ is established for arbitrary bounded domains. Relations between the weighted Bergman kernel and the Szegő kernel are also discussed.


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Additional Information

Bo-Yong Chen
Affiliation: Department of Mathematics, Tongji University, Shanghai 200092, People’s Republic of China
Email: boychen@tongji.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06113-8
PII: S 0002-9947(2014)06113-8
Keywords: $\partial-$equation, $L^2-$estimate, Bergman space, weighted Bergman kernel, Szeg\H{o} kernel
Received by editor(s): July 29, 2012
Published electronically: March 26, 2014
Additional Notes: This work was supported by the Key Program of NSFC No. 11031008
Dedicated: Dedicated to Professor Jinhao Zhang on the occasion of his seventieth birthday
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.