Titchmarsh’s convolution theorem on groups

Abstract

There is a well-known theorem of Titchrnarsh concerning measures with compact-support which may be stated as follows. (T) If ,u, v are finite measures on R (the line) with compact support, and their intervals of support are [a, b], [c, d] resp. then the interval of support of ,u * v is [a+c, b+d]. Recall that the interval of support of a measure is the smallest interval which contains the support of the measure. Note that the fact that supp(,u * v) C [a+c, b +d ] is trivial, what the theorem asserts is that supp(, * v) is contained in no smaller interval. The theorem was extended by Lions [21 to finite measures on Rn with the convex closure of the support of a mneasure replacing the interval of support. In the following, we take up the question of characterizing those locally compact abelian groups for which an analogue of (T) holds. Our proof, like Lions', relies on a reduction to the one dimensional theorem, but unlike his makes only minimal use' of complex methods, and thus is in the spirit of the real variable proof of RyllNardzewski [4] or the functional analytic proof of Kalisch [1] for (T). We begin with a very special case of (T), and investigate its validity for general l.c.a. groups G. The special case is (ST) If ,u and v are finite measures on G with compact support and A * v = O then either u = 0 or v = 0. If G is a compact group, and ,u denotes Haar mneasure, set v =I where I is any character of G; then it is easily checked that ,u * v = 0, while both are nonzero with compact support. Thus (ST) fails to hold if G is compact. In the same way one sees that (ST) doesn't hold if G contains a nontrivial compact subgroup. The remarkable fact is that if G doesn't contain any nontrivial compact subgroups (ST) holds and what is more so does an appropriate version of (T). Denote the dual of G by 'G and let Ge be the connected component of the identity in 'G. If Ge is not all of 'G there is a nontrivial subgroup H of G containing Ge such that LG/H is discrete [6]. The duality theorem gives a nontrivial compact subgroup of G; and thus G possessing no compact subgroups implies 'G connected. Recall that a character x of G is a continuous homomorphism of G into the additive