Estimates for solutions of the $\partial$-equation and application to the characterization of the zero varieties of the functions of the nevanlinna class for lineally convex domains of finite type

TL;DR

This paper extends sharp $ar ext{ extpartial}$-equation estimates to lineally convex domains of finite type and uses these results to characterize zero sets of Nevanlinna class functions in such domains.

Contribution

It introduces a new approach using a kernel-based representation formula and Kohn's $L^2$ theory for solving the $ar ext{ extpartial}$-equation in lineally convex domains.

Findings

Established sharp $ar ext{ extpartial}$-equation estimates for lineally convex domains.

Provided a characterization of zero sets of Nevanlinna class functions in these domains.

Developed a novel method replacing support function techniques with kernel-based representation.

Abstract

In the late ten years, the resolution of the equation $\bar\partial u=f$ with sharp estimates has been intensively studied for convex domains of finite type by many authors. In this paper, we consider the case of lineally convex domains. As the method used to obtain global estimates for a support function cannot be carried out in this case, we use a kernel that does not gives directly 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.