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

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 thatf^p 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.

:=:

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: H^p := { ∈L _p (T) : ƒ(z) = Σ ^∞₀ a _k z^k}; H^p 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)

Authors and Affiliations

1. Department of Mathematics, Brown University, 151 Thayer Street, Box 1917, 02912, Providence, RI, USA

   S. Treil

Corresponding author

Correspondence to S. Treil.

Partially supported by NSF grant DMS-9970395.

Reprints and permissions