Weighted Bergman spaces and the ∂̄-equation

Chen, Bo-Yong

doi:10.1090/S0002-9947-2014-06113-8

Weighted Bergman spaces and the $\bar {\partial }-$equation
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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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