Abstract
The main result of the paper is that there exist functionsf1,f2,f inH∞ satisfying the “corona condition”
such thatf 2 does not belong to the idealI generated byf 1,f 2, i.e.,f 2 cannot be represented as f2 ≡ f1g1 + f2g2, g1, g2 ∃ H∞. This gives a negative answer to an old question of T. Wolff [10].
It had been previously known under the same assumptions thatfp belongs to the ideal ifp > 2 but a counterexample can be constructed for p < 2; thus our casep = 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
such that for any g1,g2 ∃ H∞ satisfying f1g1 + f2g2 ≡ 1 (solution of the Corona Problem), the estimate ¦g1¦ ≥Cδ-2log(-log δ) holds. The estimate ¦g1¦∞ ≥Cδ-2 was obtained earlier by V. Tolokonnikov.
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Abbreviations
- :=:
-
equal by definition
- ℂ:
-
the complex plane
- D:
-
the unit disc, D := {z ∃ ℂ : ¦z¦< 1
- T:
-
unit circle
- T:
-
∂D = {z ∃ ℂ : ¦z¦ = 1
- H1, H∞ :
-
Hardy spaces: Hp := { ∈L p (T) : ƒ(z) = Σ ∞0 a k zk}; Hp can be naturally identified with the corresponding space of analytic functions on the disc D. In particular, H∞ consists of all bounded analytic functions on the unit disc D (with thesupremum norm)
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Partially supported by NSF grant DMS-9970395.
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Treil, S. Estimates in the corona theorem and ideals ofH∞: A problem of T. Wolff. J. Anal. Math. 87, 481–495 (2002). https://doi.org/10.1007/BF02868486
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DOI: https://doi.org/10.1007/BF02868486