Abstract
In this paper, we investigate the ∂-complex on weighted Bergman spaces on Hermitian manifolds satisfying a certain holomorphicity/duality condition. This generalizes the situation of the Segal-Bargmann space in Cn, studied earlier by the first-named author, in which the adjoint of the differentiation is the multiplication by z. The results are applied to two important examples in the unit ball, namely, the complex hyperbolic metric and a conformally Kähler metric which are related to Bergman spaces with so-called “exponential” and “standard” weights, respectively. In particular, we obtain new estimates for the solutions of the ∂-equation on these weighted Bergman spaces.
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