BMO estimates for the H-infinity (B-n) Corona problem

摘要

We study the H-infinity(B-n) Corona problem Sigma(N)(j=1) f(j)g(j) = h and show it is always possible to find solutions f that belong to BMOA(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-infinity, BMOA with N = infinity, while the latter result obtains BMOA(B-n) solutions for just N = 2 generators with h = 1. Our method of proof is to solve partial derivative-problems and to exploit the connection between BMO functions and Carleson measures for H-2(B-n). Key to this is the exact structure of the kernels that solve the partial derivative 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. (C) 2009 Elsevier Inc. All rights reserved.