Continuity of convolution of test functions on Lie groups

Abstract

If G is a Lie group, let D(G) be the space of compactly supported smooth functions on G. Consider the bilinear map B : D(G) x D(G) -> D(G), (f,g) |-> f*g which takes a pair of test functions to their convolution. We show that B is continuous if and only if G is sigma-compact. More generally, let r,s,t be non-negative integers (or infinity) with t <= r+s. Let E_1, E_2 be locally convex spaces and b : E_1 x E_2 -> F be a continuous bilinear map to a complete locally convex space F. The main result is a characterization of those (G,r,s,t,b) for which the convolution map C^r_c(G,E_1) x C^s_c(G,E_2) -> C^t_c(G,F) associated with b 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.