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Continuity of convolution of test functions on Lie groups |
46 pages
Published:2012-10-03
- Lidia Birth,
- Helge Glöckner,
Abstract
For a Lie group $G$, we show that the map
$C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$,
$(\gamma,\eta)\mapsto \gamma*\eta$
taking a pair of
test functions to their convolution is continuous if and only if $G$ is $\sigma$-compact.
More generally, consider $r,s,t
\in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$
and a continuous bilinear map $b\colon E_1\times E_2\to F$
to a complete locally convex space $F$.
Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$,
$(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map.
The main result is a characterization of those $(G,r,s,t,b)$
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: | Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigma-compactness, convolution, continuity, seminorm, product estimates |
| MSC Classifications: |
22E30 - Analysis on real and complex Lie groups [See also 33C80, 43-XX] 46F05 - Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35] 22D15 - Group algebras of locally compact groups 42A85 - Convolution, factorization 43A10 - Measure algebras on groups, semigroups, etc. 43A15 - $L^p$-spaces and other function spaces on groups, semigroups, etc. 46A03 - General theory of locally convex spaces 46A13 - Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40] 46E25 - Rings and algebras of continuous, differentiable or analytic functions {For Banach function algebras, see 46J10, 46J15} |