一类全纯函数的$F$ -代数中素理想的刻划
A characterization of some prime ideals in certain $F$-algebras of holomorphic functions
摘要：
类$M^p$ $(1<p<\infty)$由开单位圆盘$\Bbb D$上的所有全纯函数$f$组成，对于$$ \int_0^{2π}\left(\log^+Mf(θ)\right)^p\,\frac{dθ}{2π}<\infty, $$，其中$Mf(θ)=\sup_{0\leqslant r<1}\big\lvert f\big(re^{iθ}\big)\big \rvert$。类$M^p$配备了由$$ ρ_p(f,g)=\left(\int_0^{2π}\log^p(1+M(f-g)(θ))\, \frac{dθ}{2π}\right)^{1/p}\quad (f,g\in M^p) $$定义的度量$ρ_p$给出的拓扑成为$F$ -代数。本文考虑类$M^p$ $(1<p<\infty)$的理想结构。我们的主要结果给出了$M^p$中非稠密子集的$M^p$的素理想的一个完整刻画.作为结果，我们得到了关于Privalov类$N^p$ $(1<p<\infty)$的一个相关的Mochizuki结果.
英文摘要：
The class $M^p$ $(1<p<\infty)$ consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ for which $$ \int_0^{2π}\left(\log^+Mf(θ)\right)^p\,\frac{dθ}{2π}<\infty, $$ where $Mf(θ)=\sup_{0\leqslant r<1}\big\lvert f\big(re^{iθ}\big)\big \rvert$. The class $M^p$ equipped with the topology given by the metric $ρ_p$ defined by $$ ρ_p(f,g)=\left(\int_0^{2π}\log^p(1+M(f-g)(θ))\, \frac{dθ}{2π}\right)^{1/p}\quad (f,g\in M^p) $$ becomes an $F$-algebra. In this paper, we consider the ideal structure of the classes $M^p$ $(1<p<\infty)$. Our main result gives a complete characterization of prime ideals in $M^p$ which are not dense subsets of $M^p$. As a consequence, we obtaiin a related Mochizuki's result concerning the Privalov classes $N^p$ $(1<p<\infty)$.
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