The problem of ideals of H∞: Beyond the exponent 3/2

Abstract

The paper deals with the problem of ideals of H∞: describe increasing functions φ ≥ 0 such that for all bounded analytic functions f1, f2, ..., fn, τ in the unit disc D the condition| τ (z) | ≤ φ ((∑ | fk (z) |2)1 / 2) ∀ z ∈ D implies that τ belong to the ideal generated by f1, f2, ..., fn, i.e. that there exist bounded analytic functions g1, g2, ..., gn such that ∑k = 1n fk gk = τ. It was proved earlier by the author that the function φ (s) = s2 does not satisfy this condition. The strongest known positive result in this direction due to J. Pau states that the function φ (s) = s2 / ((ln s-1)3 / 2 ln ln s-1) works. However, there was always a suspicion that the critical exponent at ln s-1 is 1 and not 3/2. This suspicion turned out (at least partially) to be true, 3/2 indeed is not the critical exponent. The main result of the paper is that one can take for φ any function of form φ (s) = s2 ψ (ln s-2), where ψ : R+ → R+ is a bounded non-increasing function satisfying ∫0∞ ψ (x) d x 0, works. © 2007 Elsevier Inc. All rights reserved.