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

TL;DR

This paper proves that solutions to the $H^{ abla}( ext{unit ball})$ Corona problem can always be found within the BMOA space for any number of generators, improving previous results and employing $ar{ ext{d}}$-problem techniques.

Contribution

The paper establishes the existence of BMOA solutions for the Corona problem with any number of generators, extending prior work limited to fewer generators or larger domains.

Findings

Solutions exist in BMOA for any number of generators

Improves previous results by covering more generators and domains

Uses $ar{ ext{d}}$-problem solving and Carleson measure techniques

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.