  The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on
the open unit disk $\Bbb D$ for which $\lim_{r\to
1}(1-r)^{1/q}\log^+M_{\infty}(r,f)=0,$ where $M_{\infty}(r,f)=\max_{\vert
z\vert\le r}\vert f(z)\vert$ with $0<r<1$. Stoll [5, Theorem 3.2] proved that
the space $F^p$ with the topology given by the family of seminorms
$\left\{\Vert \cdot\Vert_{q,c}\right\}_{c>0}$ defined for $f\in F^q$ as $\Vert
f\Vert_{q,c}:=\sum_{n=0}^{\infty}\vert a_n\vert\exp\left(-cn^{1/(q+1)}
\right)<\infty,$ is a countably normed Fr\'{e}chet algebra. Notice that for
each $p>1$, $F^p$ is the Fr\'{e}chet envelope of the Privalov space $N^p$. In
this paper we study the structure of maximal ideals in the algebras $F^p$
($1<p<\infty$). In particular, we give a complete characterization of closed
maximal ideals in $F^p$. Moreover, we characterize multiplicative linear
functionals on $F^p$.
