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Composition of inner mappings on the ball


Authors: Jörg Eschmeier and Roland Wolff
Journal: Proc. Amer. Math. Soc. 130 (2002), 95-102
MSC (2000): Primary 32H02, 46E15
Published electronically: July 25, 2001
MathSciNet review: 1855625
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Abstract | References | Similar Articles | Additional Information

Abstract:

Suppose that $F:B_k\to B_m$ is an inner map and that $G\in H^\infty(B_m)^n$. We show that the identity

\begin{displaymath}(G\circ F)^\ast=r(G)\circ F^\ast \end{displaymath}

holds with an abstract boundary value $r(G)$. If the natural compatibility condition $\sigma_k^{F^\ast}\ll\sigma_m$ is satisfied, then $r(G)=G^\ast$. Here, $\sigma_k^{F^\ast}$denotes the image of the surface measure on $S_k$ under $F^\ast$. In particular, $G\circ F$ is inner if $F$ and $G$ are inner and $\sigma_k^{F^\ast}\ll\sigma_m$. Furthermore, we characterize the boundedness of composition operators on Hardy spaces in terms of the absolute continuity of $\sigma_k^{F^\ast}$.


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Additional Information

Jörg Eschmeier
Affiliation: Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
Email: eschmei@math.uni-sb.de

Roland Wolff
Affiliation: Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
Email: wolff@math.uni-sb.de

DOI: https://doi.org/10.1090/S0002-9939-01-06302-X
Keywords: Inner mapping, composition operator, Henkin measure, abstract boundary values
Received by editor(s): May 24, 2000
Published electronically: July 25, 2001
Communicated by: Steven R. Bell
Article copyright: © Copyright 2001 American Mathematical Society