Relation between Solvability and a Regularity of Convolution Operators in K′p, p>1

Let K′p be the space of distributions on Rn which grow no faster than ek|x|p for some k>0 and let O′c(K′p; K′p) be the space of convolution operators in K′p. We show that, for S∈o′c(K′p; K′p), S ∗ K′p=K′p is equivalent to the following: Every distribution u∈o′c(K′p; K′p) with S ∗ u∈Kp is in Kp.