Let f be a kernel in the Orlicz space LA. What is the necessary and sufficient condition on the Young's functions A, B, C so that the operator h(x) = S f(x-t)g(t) dt be compact from LB into LC? It is shown that the problem is impossible on the real line, or more generally, on a locally compact, commutative, connected but not compact group. If the group is compact, it is proved that the problem is possible, and the necessary and sufficient condition is that for every 0 > 0, there exists a number n > 0 such that for all x >/ 1: A-1 (x) B-1 (nx) /< onxc-1 (nx)