Toeplitz Corona Theorems for the Polydisk and the Unit Ball

TL;DR

This paper extends and refines existing results on the Corona problem for the polydisk and the unit ball in complex n-dimensional space, contributing to the understanding of bounded analytic functions.

Contribution

It advances the theory of the Corona problem by providing new Toeplitz corona theorems for the polydisk and the unit ball, building on prior work by Agler-McCarthy and Amar.

Findings

Extended Corona theorems for the polydisk and unit ball

Refined conditions for solvability of the Corona problem

Improved bounds in Toeplitz corona theorems

Abstract

The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in $\mathbb{C}^n$.