Vol. 4, No. 4, 2011

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ISSN: 1948-206X (e-only)
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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 Hn2 on the unit ball in 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 Bpσ has the “baby corona property” for all σ 0 and 1 < p < . 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.

Milestones
Received: 10 March 2010
Revised: 25 May 2010
Accepted: 23 June 2010
Published: 9 January 2012
Authors
Şerban Costea
Department of Mathematics and Statistics
McMaster University
1280 Main Street West
Hamilton, ON  L8S 4K1
Canada
École Polytechnique Fédérale de Lausanne
EPFL SB MATHGEOM
Station 8
CH-1015 Lausanne
Switzerland
Eric T. Sawyer
Department of Mathematics and Statistics
McMaster University
1280 Main Street West
Hamilton, ON  L8S 4K1
Canada
Brett D. Wick
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
United States