Continuity of convolution of test functions on Lie groups

TL;DR

This paper characterizes the conditions under which the convolution operation on test functions, measures, and L^1-functions on Lie groups and locally compact groups is continuous, focusing on the role of sigma-compactness and differentiability classes.

Contribution

It provides a comprehensive characterization of the continuity of convolution maps on Lie groups and locally convex spaces, extending previous results to various function spaces and measures.

Findings

Convolution is continuous on D(G) if and only if G is sigma-compact.

Characterization of continuity for convolution maps between C^r_c, C^s_c, and C^t_c spaces.

Extension of convolution continuity results to L^1-functions and Radon measures.

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.