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

The paper deals with the problem of ideals of H∞H∞: describe increasing functions φ⩾0φ⩾0 such that for all bounded analytic functions f1,f2,…,fn,τf1,f2,…,fn,τ in the unit disc DD the condition|τ(z)|⩽φ((∑|fk(z)|2)1/2)∀z∈D implies that τ   belong to the ideal generated by f1,f2,…,fnf1,f2,…,fn, i.e. that there exist bounded analytic functions g1,g2,…,gng1,g2,…,gn such that ∑k=1nfkgk=τ.It was proved earlier by the author that the function φ(s)=s2φ(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/((lns−1)3/2lnlns−1)φ(s)=s2/((lns−1)3/2lnlns−1) works. However, there was always a suspicion that the critical exponent at lns−1lns−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(lns−2)φ(s)=s2ψ(lns−2), where ψ:R+→R+ is a bounded non-increasing function satisfying ∫0∞ψ(x)dx<∞. In particular any of the functionsφ(s)=s2/((lns−2)(lnlns−2)…(lnln…ln︸mtimess−2)(lnln…ln︸m+1timess−2)1+ε),ε>0, works.