Estimates for the Corona Theorem on $H^{\infty}_{\mathbb{I}}(\D)$

TL;DR

This paper extends the corona theorem to infinitely many generators for a specific sub-algebra of bounded holomorphic functions, providing new estimates and confirming a conjecture by Ryle.

Contribution

It proves the corona theorem for infinitely many generators on $H^{I}_{I}$, a significant extension of known finite cases, and offers estimates for solutions.

Findings

Corona theorem holds for infinitely many generators in $H^{I}_{I}$.

Provides explicit estimates for corona solutions.

Confirms Ryle's conjecture and generalizes Wolff's Ideal Theorem.

Abstract

Let $\mathbb{I}$ be a proper ideal of $H^{\infty}(\D)$. We prove the corona theorem for infinitely many generators on the algebra $H^{\infty}_{\mathbb{I}}$ in which the corona theorem for finitely many functions is known to hold. This settles the conjecture of Ryle \cite{ryle1}. We also provide the estimates for corona solutions. Moreover, we prove a generalized Wolff's Ideal Theorem for this sub-algebra.