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

Tom 209 / 2012

Mohammad Ali Ardalani, Wolfgang Lusky Studia Mathematica 209 (2012), 225-234 MSC: Primary 46E15; Secondary 47B38. DOI: 10.4064/sm209-3-2


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 $Hv(\mathbb G)$.


  • Mohammad Ali ArdalaniDepartment of Mathematics
    Faculty of Science
    University of Kurdistan
    Pasdaran Ave.
    Postal code: 66177-15175
    Sanandaj, Iran
  • Wolfgang LuskyInstitute for Mathematics
    University of Paderborn
    Warburger Str. 100
    D-33098 Paderborn, Germany

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