Computational Methods and Function Theory 5 (2005), No. 2, 381--393
Copyright Heldermann Verlag 2005
Marko Kotilainen
marko.kotilainen@joensuu.fi , Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland.
Visa Latvala
visa.latvala@joensuu.fi , Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland.
Jie Xiao
jxiao@math.mun.ca , Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL, A1C5S7, Canada.
We introduce the so-called Bloch-Sobolev function spaces and show that these spaces have nice closure properties. We also characterize the boundedness and compactness of a composition operator $C_\phi$ (with analytic symbol $\phi$ between two subdomains $\Omega, \Omega'\subsetneq \mathbb{R}^2$) acting between two Bloch-Sobolev spaces. As a by-product we obtain a characterization of those analytic mappings $\phi\colon\Omega\to\Omega'$, which are uniformly continuous with respect to the quasihyperbolic metrics in $\Omega$ and $\Omega'$.
Keywords: Bloch-Sobolev spaces, composition operators, analytic mappings.
MSC 2000: Primary 31B05; Secondary 46E15, 47B38.
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