f«I,.):=

The following example shows that the acting condition K H : C(D)-+ C(D) does not imply the acting conditions K : C(D) -+ C(D) or H : C(D) -+ C(D):

(17.31)

Example 17.1. Let 9 : lIt ~ lIt be a continuous function, not identical zero, p := l(a +b),

{ g(u) if a ~ t ~ p,

h(t, s, u) := -g(u) if p < t ~ b,

let, s, r) := -1 if p < t ~ b,

and m(t, s, 0') = n(t, s, r, 0') == o. Obviously, the operators M and H do not map the space C(D) into itself, but the operator

LH:e(t,s) =16g(z(r,s»dr does.