We study estimates for the solution of the equation du=f in one variable. The new ingredient is the use of holomorphic functions with precise growth restrictions in the construction of explicit solution to the equation.
We give a sufficient condition for a hermitian holomorphic vector bundle over the disk to be quasi-isometric to the trivial bundle. One consequence is a version of Cartan's lemma on the factorization of matrices with uniform bounds.
We study the solvability of equations associated with a complex vector field in with or coefficients. We assume that is elliptic everywhere except on a simple and closed curve . We assume that, on , is of infinite type and that vanishes to a constant order. The equations considered are of the form , with satisfying compatibility conditions. We prove, in particular, that when the order of vanishing of is , the equation is solvable in the category but not in the category....