$H^\infty$的理想问题：超过指数3/2
The problem of ideals of $H^\infty$: beyond the exponent 3/2
摘要：
本文讨论了$H^\infty$的理想问题：描述增函数$ϕ\ge 0$，使得对于单位圆$D$中的所有有界解析函数$f_1,f_2,...,f_n, τ$，条件$|τ(z) | \le ϕ(\sum_k |f_k(z)|)$对所有$z\in D$，意味着$τ$属于由$f_1,f_2,...,f_n$生成的理想.作者在前面已经证明了函数$ϕ(s) =s^2$不起作用。本文的主要结果是：$ϕ$可以取任何形式为$ϕ(s) =s^2 ψ(\ln s^{-2})$的函数，其中$ψ$是$[0, \infty)$上满足$\int_0^\infty ψ(x) dx <\infty$的有界非增函数.
英文摘要：
The paper deals with the problem of ideals of $H^\infty$: describe increasing functions $ϕ\ge 0$ such that for all bounded analytic functions $f_1,f_2,...,f_n, τ$ in the unit disc $D$ the condition $|τ(z) | \le ϕ(\sum_k |f_k(z)|)$ for all $z\in D$, implies that $τ$ belong to the ideal generated by $f_1,f_2,...,f_n$. It was proved earlier by the author that the function $ϕ(s) =s^2$ does not work. The main result of the paper is that one can take for $ϕ$ any function of form $ϕ(s) =s^2 ψ(\ln s^{-2})$, where $ψ$ is a bounded non-increasing function on $[0, \infty)$ satisfying $\int_0^\infty ψ(x) dx <\infty$.
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