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Composition of inner mappings on the ball

Author(s): Jörg Eschmeier; Roland Wolff
Journal: Proc. Amer. Math. Soc. 130 (2002), 95-102.
MSC (2000): Primary 32H02, 46E15
Posted: July 25, 2001
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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: 10.1090/S0002-9939-01-06302-X
PII: S 0002-9939(01)06302-X
Keywords: Inner mapping, composition operator, Henkin measure, abstract boundary values
Received by editor(s): May 24, 2000
Posted: July 25, 2001
Communicated by: Steven R. Bell
Copyright of article: Copyright 2001, American Mathematical Society

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