Bounded operators on weighted spaces of holomorphic functions on the upper half-plane

Mohammad Ali Ardalani¹ ; Wolfgang Lusky²

¹ Department of Mathematics Faculty of Science University of Kurdistan Pasdaran Ave. Postal code: 66177-15175 Sanandaj, Iran
² Institute for Mathematics University of Paderborn Warburger Str. 100 D-33098 Paderborn, Germany

Studia Mathematica, Tome 209 (2012) no. 3, pp. 225-234

Résumé

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 $Hv(\mathbb G)$.

Détail

DOI : 10.4064/sm209-3-2

Keywords: standard weight upper half plane mathbb mathbb rightarrow mathopen infty mathclose continuous satisfies mathop mathbb geq geq lim rightarrow put mathop mathbb characterize boundedness surjectivity differentiation operator mathbb rightarrow mathbb example bounded only moderate growth study composition operators mathbb