Integral Kernels and Hölder Estimates for ∂̄ on Pseudocovnex Domains of Finite Type in C2

Abstract

In 1969 H. Grauert and I. Lieb, and, independently, G. M. Henkin, introduced methods of integral representations on strictly pseudoconvex domains in order to construct and estimate solutions of the Cauchy-Riemann equations, in analogy to the classical Cauchy transform in one complex variable. (See [R2] for a comprehensive discussion and references.) Thereafter, partial results were obtained on convex domains (see, for example, [R1], [DWF], [A]). All these results made use of rather explicit holomorphic support functions at each boundary point of the domain. But as is well known since the discovery of the famous example of Kohn and Nirenberg in 1972 [KN], such support functions do not exist in general, even if the domain is of finite type in C2, as defined by Kohn [K]. Thus new methods are required in order to find suitable integral representations on domains of finite type.