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

TL;DR

This paper constructs solution operators for the $ar ext{d}$ equation on product domains in complex space, using integral operators and solving sub-equations, with norm estimates provided.

Contribution

It introduces a method to solve the $ar ext{d}$ equation on product domains in $ extbf{C}^n$, extending known techniques to higher dimensions with new estimates.

Findings

Explicit integral operators for 1D factors

Solution via sub-$ar ext{d}$ equations for higher dimensions

Norm estimates for the constructed operators

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.