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Department of Mathematics and Statistics
of University of Helsinki
Formula 1. BlochtoBMOAcomposition operator.
Formula 2. Metric on the hyperbolic Bloch class.

Let Ω be a complex domain (in one or,
respectively, N ≥ 2
dimensions), and let φ: Ω → Ω be an analytic
mapping. The analytic composition operator with symbol φ
is by definition the mapping f → f o φ,
where f belongs to some function space on Ω.
SOME RECENT ARTICLES

O.Blasco, M.Lindström, J.Taskinen:
BlochtoBMOA compositions in several complex
variables.
Complex Var. Theory Appl. 50, 14 (2005), 10611080.
We study analytic composition operators from the Bloch space to the
analytic BMOspace; the domain of the spaces is the unit ball
of Cⁿ. Under mild regularity condition for
the symbol
φ we are able to characterize the boundedness and
compactness of the operator in terms of a Carleson measure type
condition; see Formula 1.
The work is continuation of a paper
in Can.Math.Bull. 47,2 (2004). We use a delicate modification of a well
known lacunary series argument,
which has also been used for example by Choe and Rim in their paper in
Acta Math.Hungarica vol. 72.

F.PerezGonzales,J.Rättyä,J.Taskinen:
Lipschiz continuous and compact composition operators on hyperbolic classes.
Mediterranean J.Math. 8,1 (2011), 123135.
We study hyperbolic function classes of Bloch, Dirichlet and
Qp functions. These are not linear spaces,
but we endow them with natural metrics, emerging from their definitions.
We prove the completeness of the spaces.
The definition of an analytic composition operator still makes
sense between these metric spaces. The main result of the paper
is that the Lipschitzcontinuity plays here the same role as
boundedness for usual linear operators in Banachspaces. We charaterize
the Lipschitzcontinuity of a composition operator from the
hyperbolic Bloch space into hyperbolic
Qp in terms of a formula similar to the
linear case.
We also observe that the natural concept of a compact operator
coincides here with the standard definition of completely
continuous operator in nonlinear functional analysis.
At the end of the paper there is some amusing discussion about
the choice of the metrics of the hyperbolic classes, with
related, interesting and probably not too difficult open problems.
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