Abstract
We study the H ∞ ( B n ) Corona problem ∑ j = 1 N 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 ∞ ⋅ BMOA with N = ∞ , while the latter result obtains BMOA ( B n ) 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 H 2 ( B n ) . 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.
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