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.
Categories: math.CA math.FA