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

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 n k=1 fkgk = 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) = s2ψ(ln s−2), where ψ :R+→R+ is a bounded non-increasing function satisfying ∞0 ψ(x)dx <∞. In particular any of the functions ϕ(s) = s2/ ln s−2 ln ln s−2 . . . ln ln . . . ln m t i mes s−2 ln ln . . . ln m+1 time s s−2 1+ε , ε>0, works.