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The \(\partial \)-complex on weighted Bergman spaces on Hermitian manifolds. (English) Zbl 1439.32013

In this paper, the authors study the \(\partial\)-complex on weighted Bergman spaces on Hermitian manifolds satisfying a certain holomorphicity or duality condition, which generalizes previous results on the Segal-Bargmann space \(A^2(\mathbb{C}^n,e^{-|z|^2})\). The new results can be applied to the unit ball with complex hyperbolic metric and the unit ball with a conformal Kähler metric cases. The authors develop a theory of the \(\partial\)-Neumann operator that is parallel to the classical \(\overline{\partial}\)-Neumann theory. Then they solve the \(\partial\)-equation on the weighted Bergman space and obtain new estimates for the solutions.

MSC:

32A36 Bergman spaces of functions in several complex variables
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