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Continuity of Convolution of Test Functions on Lie Groups
Published online by Cambridge University Press: 20 November 2018
Abstract
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For a Lie group 
$G$, we show that the map
$C_{c}^{\infty }\,\left( G \right)\,\times \,C_{c}^{\infty }\,\left( G \right)\,\to \,C_{c}^{\infty }\,\left( G \right),\,\left( \gamma ,\,\eta \right)\mapsto \,\gamma \,*\,\eta $
taking a pair of test functions to their convolution, is continuous if and only if $G$ is 
$\sigma -$compact. More generally, consider 
$r,\,s,\,t\,\in {{\mathbb{N}}_{0}}\,\cup \,\left\{ \infty \right\}$ with 
$t\,\le \,r\,+\,s$, locally convex spaces 
${{E}_{1}}\,,\,{{E}_{2}}$ and a continuous bilinear map 
$b:\,{{E}_{1}}\,\times \,{{E}_{2}}\,\to \,F$ to a complete locally convex space 
$F$. Let 
$\beta :\,C_{c}^{r}\,\left( G,\,{{E}_{1}} \right)\,\times \,C_{c}^{S}\,\left( G,\,{{E}_{2}} \right)\,\to$
$C_{c}^{t}\,\left( G,\,F \right),\,\left( \gamma ,\,\eta \right)\,\mapsto \,\gamma \,*\,b\,\eta$ be the associated convolution map. The main result is a characterization of those 
$\left( G,\,r,s,t,b \right)$ for which 
$\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported 
${{L}^{1}}$-functions and convolution of compactly supported Radon measures.
Keywords
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- Research Article
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- Copyright © Canadian Mathematical Society 2014
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