On $L_1$-subspaces of holomorphic functions
Volume 198 / 2010
Studia Mathematica 198 (2010), 157-175
MSC: Primary 46E15; Secondary 46B03.
DOI: 10.4064/sm198-2-4
Abstract
We study the spaces \[ H_{ \mu}( \Omega) = \bigg\{ f: \Omega \rightarrow \mathbb C \hbox{ holomorphic}:\int_0^R \int_0^{2 \pi} |f(re^{i \varphi})| \,d \varphi \,d \mu(r) < \infty \bigg\} \] where $ \Omega$ is a disc with radius $R$ and $ \mu$ is a given probability measure on $[0,R[$. We show that, depending on $ \mu$, $H_{ \mu}( \Omega)$ is either isomorphic to $l_1$ or to $ \left( \sum \oplus A_n \right)_{(1)}$. Here $A_n$ is the space of all polynomials of degree $ \leq n$ endowed with the $L_1$-norm on the unit sphere.
