#### Vol. 4, No. 4, 2011

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

### Şerban Costea, Eric T. Sawyer and Brett D. Wick

Vol. 4 (2011), No. 4, 499–550
##### 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. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.

##### Keywords
Besov–Sobolev Spaces, corona Theorem, several complex variables, Toeplitz corona theorem
##### Mathematical Subject Classification 2000
Primary: 30H05, 32A37
##### Supplementary material

PDF file containing proofs of formulas and modifications of arguments already in the literature that would otherwise interrupt the main flow of the paper.