Canonical Homotopy Operators for @ in the Ball with Respect to the Bergman Metric

Abstract

We notice that some well-known homotopy operators due to Skoda et. al. for the $\bar\partial$-complex in the ball actually give the boundary values of the canonical homotopy operators with respect to certain weighted Bergman metrics. We provide explicit formulas even for the interior values of these operators. The construction is based on a technique of representing a $\bar\partial$-equation as a $\bar\partial_b$-equation on the boundary of the ball in a higher dimension. The kernel corresponding to the operator that is canonical with respect to the Euclidean metric was previously found by Harvey and Polking. Contrary to the Euclidean case, any form which is smooth up to the boundary belongs to the domain of the corresponding operator $\bar\partial^*$, with respect to the metrics we consider. We also discuss the corresponding $\bar\square$-operator and its canonical solution operator.

Moreover, our homotopy operators satisfy a certain commutation rule with the Lie derivative with respect to the vector fields $\partial/\partial\zeta_k$, which makes it possible to construct homotopy formulas even for the $\partial\bar\partial$-operator.