The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in $\mathbb{C}^{n}$

TL;DR

This paper extends the Corona Theorem to the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in complex space, establishing new corona properties and generalizations.

Contribution

It proves the Corona Theorem for the multiplier algebra of the Drury-Arveson space and introduces the baby corona property for Besov-Sobolev spaces, generalizing classical results.

Findings

No corona in the maximal ideal space of the multiplier algebra.

The Besov-Sobolev space $B_{p}^{\sigma}$ has the baby corona property.

Infinite generator and semi-infinite matrix versions of the corona theorems.

Abstract

We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\mathbb{C}^{n}$ has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^{\sigma}$ has the "baby corona property" for all $\sigma \geq 0$ and $1<p<\infty $. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.