BMO estimates for the H∞(Bn) Corona problem

We study the H∞(Bn) Corona problem 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.