On the boundedness of the differentiation operator between weighted spaces of holomorphic functions

Anahit Harutyunyan¹ ; Wolfgang Lusky²

¹Faculty for Informatics and Applied Mathematics University of Yerevan Alek Manukian 1 Yerevan 25, Armenia
²Institute for Mathematics University of Paderborn Warburger Str. 100 D-33098 Paderborn, Germany

Studia Mathematica, Tome 184 (2008) no. 3, pp. 233-247

Résumé

We give necessary and sufficient conditions on the weights $v$ and $w$ such that the differentiation operator $D: Hv( \Omega) \rightarrow Hw( \Omega)$ between two weighted spaces of holomorphic functions is bounded and onto. Here $ \Omega = \mathbb C$ or $ \Omega = \mathbb D$. In particular we characterize all weights $v$ such that $ D:Hv( \Omega) \rightarrow Hw( \Omega)$ is bounded and onto where $w(r) = v(r)(1-r)$ if $ \Omega = \mathbb D$ and $w=v$ if $ \Omega = \mathbb C$. This leads to a new description of normal weights.

Détail

DOI : 10.4064/sm184-3-3

Keywords: necessary sufficient conditions weights differentiation operator omega rightarrow omega between weighted spaces holomorphic functions bounded here omega mathbb omega mathbb particular characterize weights omega rightarrow omega bounded where r omega mathbb omega mathbb leads description normal weights