Equivalency of the Corona problem and Gleason problem in the theory of SCV

TL;DR

This paper proves the equivalence of the Corona and Gleason problems in several complex variables and applies this to solve both problems affirmatively for various domains and function spaces.

Contribution

It establishes the equivalency between the Corona and Gleason problems and extends solutions to multiple complex domains and Banach spaces of holomorphic functions.

Findings

Confirmed the equivalency of the Corona and Gleason problems.

Provided affirmative solutions for the problems on various domains.

Extended results to Banach spaces of holomorphic functions.

Abstract

We establish an equivalency of the Corona problem (1962) and Gleason problem (1964) in the theory of several complex variables. As an application, we give an affirmative solution of the Corona problem for certain bounded pseudoconvex domains or polydomains including balls and polydiscs. Indeed, we extend our recent work on Gleason problem based on the functional analytic approach, as well as extend recent work of Clos. We also use this equivalency or else other (functional analytic) methods to affirmatively solve both problems for various Banach spaces of bounded holomorphic functions (including certain holomorphic mixed-norm spaces) on various types of domains such as holomorphic Holder and Lipschitz spaces (left open by Fornaess and Ovrelid in 1983), holomorphic mean Besov-Lipschitz spaces, Besov-Lipschitz spaces, Hardy-Sobolev spaces, and a weighted Bergman space. The discussion goes via first studying Lipschitz algebras of holomorphic functions of order a, where a in (0, 1]; in particular, the Gelfand theory and the maximal ideal spaces of these algebras are discussed.