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

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

v1: 70 pgs; v2: 73 pgs.; introduction expanded, clarified. v3: 73 pgs.; restriction in main result removed (see 9.2), arguments expanded (see 8.1.1). v4: 74 pgs.; systematic arithmetic misprints fixed on pgs. 37-48. v5: 76 pgs.; incorrect embedding corrected via Proposition 4. v6: 80 pgs.; main result extended to vector-valued setting. v7: 82 pgs.; typos removed.

References in corpus (3)

• The Matrix-Valued $H^p$ Corona Problem in the Disk and Polydisk
• Toeplitz Corona Theorems for the Polydisk and the Unit Ball
• BMO Estimates for the $H^{\infty }({\mathbb{B}}_n)$ Corona Problem

Cited by in corpus (8)

• The Smirnov class for spaces with the complete Pick property
• Factorizations induced by complete Nevanlinna-Pick factors
• Spaces of Dirichlet series with the complete Pick property
• Paley--Wiener theorems on the Siegel upper half-space
• Two classes of integral operators over the Siegel upper half-space
• Henkin measures for the Drury-Arveson space
• Ideal membership in $H^{\infty }$: Toeplitz corona approach
• The Corona Problem for Kernel Multiplier Algebras