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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Weighted Bergman spaces and the $\bar {\partial }-$equation
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by Bo-Yong Chen PDF
Trans. Amer. Math. Soc. 366 (2014), 4127-4150 Request permission

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
  • 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
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 4127-4150
  • MSC (2010): Primary 32A25, 32A36, 32A40, 32W05
  • DOI: https://doi.org/10.1090/S0002-9947-2014-06113-8
  • MathSciNet review: 3206454