BMO Estimates for the $H^{\infty }({\mathbb{B}}_n)$ Corona Problem

Abstract

We study the $H^{\infty }({\mathbb{B}}_n)$ Corona problem ${\sum }_{j=1}^N{f}_j{g}_j=h$ and show it is always possible to find solutions $f$ that belong to $BMOA({\mathbb{B}}_n)$ for any $n>1$, including infinitely many generators $N$. This theorem improves upon both a 2000 result of Andersson and Carlsson and the classical 1977 result of Varopoulos. The former result obtains solutions for strictly pseudoconvex domains in the larger space $H^{\infty }\cdot BMOA$ with $N=\infty$, while the latter result obtains $BMOA({\mathbb{B}}_n)$ solutions for just N=2 generators with $h=1$. Our method of proof is to solve $\overline{\partial }$-problems and to exploit the connection between $BMO$ functions and Carleson measures for $H^2({\mathbb{B}}_n)$. Key to this is the exact structure of the kernels that solve the $\overline{\partial }$ equation for $(0,q)$ forms, as well as new estimates for iterates of these operators. A generalization to multiplier algebras of Besov-Sobolev spaces is also given.

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Cited by in corpus (1)

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