\(\overline\partial\) and Schrödinger operators (Q1922574)

We study the canonical, or \(L^2\)-minimal, solution to the \(\overline\partial\)-equation in weighted \(L^2\)-spaces in domains in \(\mathbb{C}\). It is shown that the canonical solution operator can be obtained from the Green's function for a certain Schrödinger operator. Most of the known facts about the one-dimensional \(\overline\partial\)-equation, including Hörmanders \(L^2\)-estimates, uniform estimates for Carleson-measure data, T. Wolff's theorem then follow from an inequality related to Kato's inequality, and sometimes in a sharper form. At the end we discuss some estimates for the asymptotic decay of the kernels which do not follow from Kato's inequality, and indicate the relation to the presence of a magnetic field in the Schrödinger operator.