The problem of ideals of $H^{\infty }$: beyond the exponent 3/2

Abstract

The paper deals with the problem of ideals of $H^{\infty }$: describe increasing functions $\varphi \ge 0$ such that for all bounded analytic functions $f_1,f_2,...,f_n,\tau$ in the unit disc $D$ the condition $|\tau (z)|\le \varphi ({\sum }_k|f_k(z)|)$ for all $z\in D$, implies that $\tau$ belong to the ideal generated by $f_1,f_2,...,f_n$. It was proved earlier by the author that the function $\varphi (s)=s^2$ does not work. The main result of the paper is that one can take for $\varphi$ any function of form $\varphi (s)=s^2\psi (\ln s^{-2})$, where $\psi$ is a bounded non-increasing function on $[0,\infty )$ satisfying ${\int }_0^{\infty }\psi (x)dx<\infty$.

17 pages

Cited by in corpus (3)

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