A characterization of some prime ideals in certain $F$-algebras of holomorphic functions

TL;DR

This paper characterizes prime ideals in certain $F$-algebras of holomorphic functions, specifically the classes $M^p$, and relates these findings to Privalov classes $N^p$, advancing the understanding of their ideal structures.

Contribution

It provides a complete characterization of prime ideals in the $F$-algebras $M^p$, extending the understanding of their algebraic structure and connecting to Mochizuki's results on $N^p$.

Findings

Complete characterization of prime ideals in $M^p$

Identification of non-dense prime ideals

Extension of results to Privalov classes $N^p$

Abstract

The class $M^p$ $(1<p<\infty)$ consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ for which $$ \int_0^{2\pi}\left(\log^+Mf(\theta)\right)^p\,\frac{d\theta}{2\pi}<\infty, $$ where $Mf(\theta)=\sup_{0\leqslant r<1}\big\lvert f\big(re^{i\theta}\big)\big \rvert$. The class $M^p$ equipped with the topology given by the metric $\rho_p$ defined by $$ \rho_p(f,g)=\left(\int_0^{2\pi}\log^p(1+M(f-g)(\theta))\, \frac{d\theta}{2\pi}\right)^{1/p}\quad (f,g\in M^p) $$ 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 $N^p$ $(1<p<\infty)$.