Product domains, Multi-Cauchy transforms, and the $\bar{\partial }$ equation

Abstract

Solution operators for the equation $\bar{\partial }u=f$ are constructed on general product domains in ${\mathbb{C}}^n$. When the factors are one-dimensional, the operator is a simple integral operator: it involves specific derivatives of $f$ integrated against iterated Cauchy kernels. For higher dimensional factors, the solution is constructed by solving sub-$\bar{\partial }$ equations with modified data on the factors. Estimates of the operators in several norms are proved.

31 pages