Bounded operators on weighted spaces of holomorphic functions on the upper half-plane
Volume 209 / 2012
Studia Mathematica 209 (2012), 225-234
MSC: Primary 46E15; Secondary 47B38.
DOI: 10.4064/sm209-3-2
Abstract
Let $v$ be a standard weight on the upper half-plane $ \mathbb G$, i.e. $v: \mathbb G \rightarrow \mathopen]0, \infty\mathclose[$ is continuous and satisfies $v(w) = v( i \mathop{\rm Im} w)$, $ w \in \mathbb G$, $v(it) \geq v(is)$ if $ t \geq s > 0$ and $ \lim_{t \rightarrow 0} v(it) = 0$. Put $v_1(w) = \mathop{\rm Im} w \, v(w)$, $ w \in \mathbb G$. We characterize boundedness and surjectivity of the differentiation operator $D: Hv(\mathbb G) \rightarrow Hv_1(\mathbb G)$. For example we show that $D$ is bounded if and only if $v$ is at most of moderate growth. We also study composition operators on $Hv(\mathbb G)$.
