### On a translation property of positive definite functions

If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant ${C}_{h}>0$ such that $\int {L}_{x}h\xb7g\le {C}_{h}\int hg$ for every continuous positive definite g≥0, where ${L}_{x}$ is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at all. We fill...