CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

Continuity of convolution of test functions on Lie groups

  Published:2012-10-03
 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
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   LaTeX   MathJax   PDF  

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 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, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25 show english descriptions Analysis on real and complex Lie groups [See also 33C80, 43-XX]
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, 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}
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/