http://hdl.handle.net/2117/3148 2013-05-25T02:13:23Z http://hdl.handle.net/2117/10512 Title: In front of the sea Authors: Bru Bistuer, Eduard 2010-12-09T16:49:35Z http://hdl.handle.net/2117/10511 Title: Universitat Politècnica de Catalunya: Escola Tècnica Superior d’Arquitectura de Barcelona Authors: Bru Bistuer, Eduard 2010-12-09T16:33:22Z http://hdl.handle.net/2117/792 Title: Canonical Homotopy Operators for @ in the Ball with Respect to the Bergman Metric Authors: Andersson, Mats; Ortega Cerdà, Joaquim 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. 2007-04-27T18:30:39Z