Sobolev space estimates for solutions of the equation \({\overline{\partial}u} =f\) on polycylinders (Q1774715)

Let \(\Omega= U_1\times\dots \times U_n\) be an admissible polycylinder (i.e., each \(U_j\) is a bounded open subset of \(\mathbb C\) with the boundary of plane measure zero). Main result: There exists a positive constant \(\delta\) such that for every \(\overline {\partial }\)-closed form \(f\in W^{k,p}_{(\gamma, q+1)}(\Omega)\) with \(1\leq p\leq \infty\), \(k=1, 2, \dots \) there exists \(u\in W^{k,p}_{(\gamma,q)}(\Omega)\) such that \(\bar {\partial}u = f\) and \[ \| u\| _{W^{k,p}_{(\gamma, q)}(\Omega)}\leq \delta\| f\| _{W^{p,q}_{(\gamma, q+1)}(\Omega)}. \] The result is applied to give a solution of the Sobolev-Corona problem and an improvement of Weinstock's approximation theorem for polycylinders.