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

The class $M^p$ $(1<p<\mathrm{\infty })$ consists of all holomorphic functions $f$ on the open unit disk $\mathbb{D}$ for which $${\int }_0^{2\pi }{({\log }^{+}Mf(\theta ))}^p\frac{d\theta }{2\pi }<\mathrm{\infty },$$ where $Mf(\theta)=\sup_{0\leqslant r<1}\big\lvert f\big(re^{i\theta}\big)\big \rvert$\mathrm{.}Theclass$M^p$equippedwiththetopologygivenbythemetric$\rho_p$definedby$$ \rho_p(f,g)=\left(\int_0^{2\pi}\log^p(1+M(f-g)(\theta)), \frac{d\theta}{2\pi}\right)^{1/p}\quad (f,g\in M^p) $$becomesan$F$-algebra. In this paper, we consider the ideal structure of the classes $M^p$ $(1<p<\mathrm{\infty })$. 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 $N^p$ $(1<p<\mathrm{\infty })$.