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

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 $\sigma \ge 0$ and $1<p<\mathrm{\infty }$. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.