Sup-norm estimates for \({\bar\partial}\) in \({\mathbb{C}}^ 2\)
This is the first really deep result about uniform estimates for the \({\bar \partial}\)-problem in fifteen years. While the overall method - that of constructing an integral formula using the Henkin method - is not new, the specific method for constructing the function \(\Phi\) (z,\(\xi)\) needed for the Cauchy-Fantappié formalism is very new and is important. In particular, unlike previous constructions, \(\Phi\) (z,\(\xi)\) is constructed microlocally; such a delicate construction is necessary because the domains in question do not necessarily have local holomorphic separating functions. The paper relies on earlier deep work of Bedford and the author about peak functions on finite type domains in \({\mathbb{C}}^ 2\). Both that and the present one are delicate and are hard work to read. But they should prove to be valuable tools in future analysis.
uniform estimates for the \({\bar \partial }\)- problem
scientific article; zbMATH DE number 3946612