Weighted Bergman spaces and the $\bar {\partial }-$equation
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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.References
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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
- © 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
Dedicated: Dedicated to Professor Jinhao Zhang on the occasion of his seventieth birthday