BMO estimates for the H∞(Bn) Corona problem

We study the H∞(Bn) Corona problem ∑j=1Nfjgj=h and show it is always possible to find solutions f that belong to BMOA(Bn) 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∞⋅BMOA with N=∞, while the latter result obtains BMOA(Bn) solutions for just N=2 generators with h=1. Our method of proof is to solve ∂¯-problems and to exploit the connection between BMO functions and Carleson measures for H2(Bn). Key to this is the exact structure of the kernels that solve the ∂¯ 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.