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A LIOUVILLE THEOREM FOR MATRIX-VALUED HARMONIC FUNCTIONS ON NILPOTENT GROUPS

Published online by Cambridge University Press:  14 August 2003

CHO-HO CHU  and
TUAN GIAO VU
CHO-HO CHU
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS [email protected]
TUAN GIAO VU
Affiliation:
Goldsmiths College, University of London, London SE14 6NW [email protected]
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Abstract

Let $\sigma$ be a non-degenerate positive $M_n$-valued measure on a locally compact group $G$ with $\|\sigma\|\,{=}\,1$. An $M_n$-valued Borel function $f$ on $G$ is called $\sigma$-harmonic if $f(x) = \int_G f(xy^{-1})\,d\sigma(y)$ for all $x\,{\in}\,G$. Given such a function $f$ which is bounded and left uniformly continuous on $G$, it is shown that every central element in $G$ is a period of $f$. Further, it is shown that $f$ is constant if $G$ is nilpotent or central.

Keywords

31C0543A0545E1046G10
Type
Notes and Papers
Information
Bulletin of the London Mathematical Society , Volume 35 , Issue 5 , September 2003 , pp. 651 - 658
Copyright
© The London Mathematical Society 2003

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