The corona theorem for the Drury–Arveson Hardy space and other holomorphic Besov–Sobolev spaces on the unit ball in $\C^n$

Abstract

We prove that the multiplier algebra of the Drury–Arveson Hardy space $H_n^2$ on the unit ball in ${ℂ}^n$ has no corona in its maximal ideal space, thus generalizing the 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 $\sigma \ge 0$ and $1<p<\infty$. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.