Hölder estimates for the $$\bar \partial - equation$$ in some domains of finite typein some domains of finite type

Abstract

We obtain Holder estimates for the\(\bar \partial - equation\) on some domains of finite type in ℂn using proper mapping techniques. The domains considered are domains of finite type in the sense of D’Angelo and are defined by local coordinate expressions satisfying certain algebraic geometric conditions which prevent the existence of complex analytic varieties in the boundary of the domain. Using a proper mapping which is given by the finite type condition and which carries all the information about the intrinsic geometry of the boundary, we transform the finite type points into strongly pseudoconvex ones. At these strongly pseudoconvex points we compute an explicit solution using the Henkin integral formula and we obtain estimates that we are able to pull back to the original domain. We achieve this by exploiting the branching behavior of the proper mapping. We also construct some biholomorphic numerical invariants associated with some of the domains under consideration.