On the isomorphism classes of weighted spaces of harmonic and holomorphic functions
Volume 175 / 2006
Abstract
Let $ {\mit\Omega}$ be either the complex plane or the open unit disc. We completely determine the isomorphism classes of \[ Hv = \{ f: {\mit\Omega} \rightarrow \mathbb C \mbox{holomorphic}: \sup_{z \in {\mit\Omega}} |f(z)|v(z) < \infty \} \] and investigate some isomorphism classes of \[ hv = \{ f: {\mit\Omega} \rightarrow \mathbb C \mbox{ harmonic} : \sup_{z \in {\mit\Omega}} |f(z)|v(z) < \infty \} \] where $v$ is a given radial weight function. Our main results show that, without any further condition on $v$, there are only two possibilities for $Hv$, namely either $Hv \sim l_{ \infty}$ or $ Hv \sim H_{ \infty}$, and at least two possibilities for $hv$, again $hv \sim l_{ \infty}$ and $hv \sim H_{ \infty}$. We also discuss many new examples of weights.
