On an $F$-algebra of holomorphic functions on the upper half plane

Abstract

In this paper, we shall consider the class $N^p(D)(p>1)$ of holomorphic functions on the upper half plane $D:=\{ z \in {\bf C} \, | \, \verb|Im| z > 0 \}$ satisfying $\displaystyle \sup_{y>0} \int_{\bf R} \Bigl( \log (1+|f(x+iy)|) \Bigr)^p \,dx < \infty$. We shall prove that $N^p(D)$ is an $F$-algebra with respect to a natural metric on $N^p(D)$. Moreover, a canonical factorization theorem for $N^p(D)$ will be given.