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A characterization of some prime ideals in certain $F$-algebras of holomorphic functions (1902.09616v1)

Published 21 Feb 2019 in math.CV

Abstract: The class $Mp$ $(1<p<\infty)$ 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 $Mp$ equipped with the topology given by the metric $\rho_p$ defined by $$ \rho_p(f,g)=\left(\int_0{2\pi}\logp(1+M(f-g)(\theta))\, \frac{d\theta}{2\pi}\right){1/p}\quad (f,g\in Mp) $$ becomes an $F$-algebra. In this paper, we consider the ideal structure of the classes $Mp$ $(1<p<\infty)$. Our main result gives a complete characterization of prime ideals in $Mp$ which are not dense subsets of $Mp$. As a consequence, we obtaiin a related Mochizuki's result concerning the Privalov classes $Np$ $(1<p<\infty)$.

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