Abstract view
Continuity of convolution of test functions on Lie groups


Published:20121003
Printed: Feb 2014
Lidia Birth,
Universität Paderborn, Institut für Mathematik,, Warburger Str. 100, 33098 Paderborn, Germany
Helge Glöckner,
Universität Paderborn, Institut für Mathematik,, Warburger Str. 100, 33098 Paderborn, Germany
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, sigmacompactness, convolution, continuity, seminorm, product estimates
Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigmacompactness, convolution, continuity, seminorm, product estimates

MSC Classifications: 
22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25 show english descriptions
Analysis on real and complex Lie groups [See also 33C80, 43XX] Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35] Group algebras of locally compact groups Convolution, factorization Measure algebras on groups, semigroups, etc. $L^p$spaces and other function spaces on groups, semigroups, etc. General theory of locally convex spaces Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40] Rings and algebras of continuous, differentiable or analytic functions {For Banach function algebras, see 46J10, 46J15}
22E30  Analysis on real and complex Lie groups [See also 33C80, 43XX] 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}
