CANONICAL HOMOTOPY OPERATORS FOR $\bar\partial$ IN THE BALL WITH RESPECT 
TO THE BERGMAN METRIC

Mats Andersson (1) and Joaquim Ortega-Cerda (2)

(1) Dept. of Mathematics, Chalmers University of Technology and Goteborg 
    University, S-412 96 Goteborg (Sweden).
    E-mail: matsa@math.chalmers.se

(2) Dept. de Matematica Aplicada I (ETSEIB), Universitat Politecnica de 
    Catalunya, Diagonal 647, 08028 Barcelona (Spain).
    E-mail: jortega@ma1.upc.es 

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. 

Keywords: Bergman metric, $\bar\partial$-equation, $\bar\partial$-Neumann 
equation, homotopy operator, integral formula.

AMS Subject Classifications: 32A25, 32F20, 35N15