Continuity of Convolution of Test Functions on Lie Groups

Birth, Lidia ; Glöckner, Helge

Canadian journal of mathematics, Tome 66 (2014) no. 1, pp. 102-140

Résumé

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.

Détail

DOI : 10.4153/CJM-2012-035-6

Mots-clés : 22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25, Lie group, locally compact group, smooth function, compact support, test function, secondcountability, countable basis, sigma-compactness, convolution, continuity, seminorm, product estimates