ABSTRACT
By construction, every operator P E £p(C) is a sum of four operators, namely the multiplication operator (13.3), the partial integral operators (13.4) and (13.5), and the ordinary integral operator (13.6). It is natural to expect that, if such an operator P acts in the space C(D), also all its components (13.3) - (13.6) act in the space C(D). Surprisingly, this is not true, as is shown by the following
Example 13.1. Consider the decomposition of the square D .- [0,1] x [0,1] into the four subsquares D 1 := [0, l] x [0, i], D2 :=a, 1] x [0, il, D3 := [0, il x (i, 1], and D4 := (!' 1] x (!, 1], and define P : C(D) -+ C(D) by
A straightforward calculation shows that P E £,,(C) with
if (t,s)EDIUD3UD4' if ! < r ~ t ~ 1 and 0 ~ s ~ !, if ! ~ t < r ~ 1 and 0 ~ s ~ i, if (t,s)ED1 UD2 UD4 , if l < u ~ s ~ 1 and 0 ~ t ~ i, if ! ~ s < u ~ 1 and 0 ~ t ~ !,
and
0 if (t, 8) E D1 U D2 U D3 , net, 8, T, 0') = O(2t-l~2.-1) if! < T ~ t ~ 1, ! < 0' ~ 8 ~ 1,
otherwise.
