A Berndtsson–Andersson operator solving $${\overline{\partial}}$$ -equation with W α -estimates on convex domains of finite type

Abstract

The Carleson measures were first introduced by Carleson in order to solve the corona problem for the disc in \({\mathbb{C}}\). The notion of Carleson measure can be generalized to any homogeneous space and were also used in the context of the corona problem in \({\mathbb{C}^n}\) for example by Varopoulos, Amar and Andersson and Carlsson. One of the steps to solve the Hp corona problem in a pseudoconvex domain is to solve the \({\overline\partial}\) equation for a form μ satisfying a Carleson condition and get norm estimates of the solution in term of the Carleson norm of μ. The main goal of this paper is to consider this question in the case of convex domains of finite type and to get estimates linked to the multitype of the domain.