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

Abstract

The class $M^p$ $(1<p<\infty )$ consists of all holomorphic functions $f$ on the open unit disk $\mathbb{D}$ for which $${\int }_0^{2\pi }{\left({\log }^{+}Mf(\theta )\right)}^p\frac{d\theta }{2\pi }<\infty ,$$ where $Mf(\theta )={\sup }_{0⩽r<1}\left|f\right(r{e}^{i\theta }\left)\right|$. The class $M^p$ equipped with the topology given by the metric ${\rho }_p$ defined by $${\rho }_p(f,g)={\left({\int }_0^{2\pi }{\log }^p(1+M(f-g)(\theta ))\frac{d\theta }{2\pi }\right)}^{1/p}(f,g\in M^p)$$ 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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Cited by in corpus (1)

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