Estimates in the corona theorem and ideals of H∞: A problem of T. Wolff

Abstract

The main result of the paper is that there exist functions f1, f2, f in H∞ satisfying the "corona condition" |f1(z)| + |f2(z)| ≥ |f(z)|, z ∈ double-struck D sign, such that f2 does not belong to the ideal I generated by f1, f2, i.e., f2 cannot be represented as f2 ≡ f1g1 + f2g2, g1, g2 ∈ H∞. This gives a negative answer to an old question of T. Wolff. It had been previously known under the same assumptions that fp belongs to the ideal if p > 2 but a counterexample can be constructed for p < 2; thus our case p = 2 is the critical one. To get the main result, we improve lower estimates for the solution of the Corona Problem. Specifically, we prove that given δ > 0, there exist finite Blaschke products f1, f2 satisfying the corona condition |f1(z)| + |f2(z)| ≥ δ, z ∈ double-struck D sign, such that for any g1, g2 ∈ H∞ satisfying f1g1 + f2g2 ≡ 1 (solution of the Corona Problem), the estimate ∥g1∥∞ ≥ Cδ-2 log(-log δ) holds. The estimate ∥;g1∥∞ ≥ Cδ-2 was obtained earlier by V. Tolokonnikov.