Sobolev space estimates for solutions of the equation ∂¯u=f on polycylinders

In trying to improve Weinstock's results on approximation by holomorphic functions on certain product domains, we are led to estimates in Sobolev spaces for the ∂-operator on polycylinders for (γ, q)-forms. This generalizes our results for the same operator on poly-cylinders previously obtained, and can be applied to a number of other problems such as the Corona problem. 1. Introduction. Had we the uniform estimates of Grauert and Lieb for the Cauchy-Riemann equation [5] on polycylinders, we would have improved Weinstock's result [6, Theorem 1.1] on polycylinders a long time ago. On the other hand, we have had several estimates for the ∂-operator in Sobolev spaces [2, 3, 4] on polycylinders, which we have been trying to improve. At the same time, we noticed that because some Sobolev norms dominate the uniform norm, if we did the approximation in those Sobolev norms, we would get the desired improvement of the above-mentioned theorem of Weinstock on polycylinders. We therefore went ahead here and jazzed up our previous estimates to get new estimates. We also applied our results to solve the Sobolev-Corona problem.