Solvability of the Gleason problem on a class of bounded pseudoconvex domains

TL;DR

This paper investigates conditions under which the Gleason problem is solvable on certain bounded pseudoconvex domains, linking it to the finite generation of ideals of bounded holomorphic functions and exploring smooth and Bergman space cases.

Contribution

It establishes that solvability of the bounded ar problem implies finite generation of ideals, extending results to smooth domains and Bergman spaces.

Findings

Finite generation of ideals when ar problem is solvable

Extension of results to smooth boundary domains

Analysis within Bergman space context

Abstract

We show that if a bounded pseudoconvex domain satisfies the solvability of the bounded $\bar{\partial}$ problem, then the ideal of bounded holomorphic functions vanishing at a point in the domain is finitely generated. We also prove a smooth analog of the main result for bounded pseudoconvex domains with a sufficiently smooth boundary and also consider the Bergman space case.