Estimates in the Corona Theorem and Ideals of H∞: a Problem of T. Wolff
نویسنده
چکیده
|f1(z)| + |f2(z)| ≥ |f(z)| ∀z ∈ D, and 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 by T. Wolff. Note, that it was well known before that under the same assumptions f belongs to the ideal if p > 2, but a counterexample can be constructed for p < 2, so our case p = 2 is a critical one. To get the main result we improved lower estimates for the solution of the Corona problem. Namely, we proved that given δ > 0 there exist finite Blaschke products f1, f2 satisfying the Corona condition |f1(z)| + |f2(z)| ≥ δ ∀z ∈ D, and 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.
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