The {bar{partial}} -Neumann problem on product domains in {mathbb{C}^{n}}

Let ${Omega=Omega_{1}timescdotstimesOmega_{n}subsetmathbb{C}^{n}}$ , where ${Omega_{j}subsetmathbb{C}}$ is a bounded domain with smooth boundary. We study the solution operator to the ${overlinepartial}$ -Neumann problem for (0,1)-forms on Ω. In particular, we construct singular functions which describe the singular behavior of the solution. As a corollary our results carry over to the ${overlinepartial}$ -Neumann problem for (0,q)-forms. Despite the singularities, we show that the canonical solution to the ${overlinepartial}$ -equation, obtained from the Neumann operator, does not exhibit singularities when given smooth data.