Estimates for Solutions of the $\bar{\partial}$ -Equation and Application to the Characterization of the Zero Varieties of the Functions of the Nevanlinna Class for Lineally Convex Domains of Finite Type

Abstract

In the last ten years, the resolution of the equation $\bar{\partial}u=f$ with sharp estimates has been intensively studied for convex domains of finite type in $\mathbb{C}^{n}$ by many authors. Generally, they used kernels constructed with holomorphic support function satisfying “good” global estimates. In this paper, we consider the case of lineally convex domains. Unfortunately, the method used to obtain global estimates for the support function cannot be carried out in that case. Then we use a kernel that does not directly give a solution of the $\bar{\partial}$ -equation, but only a representation formula which allows us to end the resolution of the equation using Kohn’s L 2 theory. As an application, we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.