ABSTRACT
Theorem 13.1. The linear operator (13.2) acts in the space C(D) if and only if the functions (13.16) - (13.19) are continuous on D for all fized (e, 11), and the function (13.20) is bounded on D. In this case the operator (13.2) is automatically bounded on C(D), and its norm is given by
o Suppose first that the operator (13.2) acts in C(D). It is known that this operator is then automatically bounded (KALITvIN-JANKELEVICH [1994]) and, by Radon's theorem, admits a representation
(13.22) Pz(t,s) = L61~ z(r,u)dg(t,&,r,u), where the function 9 is given by (13.15) and has the property that g(t, &, ., .) is of bounded variation and g( ., ., r, (1) is weakly continuous. The weak continuity of g(.,., r, u) is equivalent, as we have seen, to the continuity of the functions (13.16) - (13.19). On the other hand, the boundedness of the function (13.20) follows from the fact that get, &,.,.) is of bounded variationj we claim that (13.23) VarDg(t,s,.,·) = ret,s). To prove (13.23), let (t, 8) be a fixed interior point of D, choose points 1£ E (a,t), v E (t,b), P E (e,s), and q E (&,d), and split the rectangle D = [a, b) x [e, d) into the 9 parts D1 := [a, 1£] X [e,p], D2 := [a, 1£] X [P, q], D3 := [a, 1£] X [q, d), D.. := [1£, v] X [q, d), Dr, := [1£, v] X [P, q], D6 := [u,v] X [e,p], D7 := [v,h] x [q,d], Ds := [",h] x [p,q], and
(13.24)
Dg := [v, b) x [c,p], Then the total variation of get, s,',·) on D is equal to the sum of the variations of get, s,.,·) on D; (j = 1, ... ,9). Now, calculating the variation of g(t,s,',.) on D; (j = 1, .. ,,9) for u, v - t and p, q _ s we get for the total variation the formula
+1"l' In(t,s,T,U)!dudT, which proves (13.23). To see in turn that (13.24) is true consider, for example, the "center rectangle" Dr> = [u, v] X (P, q]. On this rectangle we have
+LtT m(t,s,iT)diT~(t,T)+ 1'"LtT n(t,s,T,iT)diTdT. Given ~ > 0, choose a partition II of Dr> such that the corresponding variation of get, s,', ,) with respect to II is less than~. By the absolute continuity of the integral and the definition of the functions ", we get IVarng(t,s, "')-lc(t,s)11 < ~,andhence IVarD,g(t,s",·)-lc(t,s)1I < ~ for u, v - t and p, q - s. The other eight equalities in (13.24) are proved similarly. Conversely, suppose now tha.t the functions (13,16) - (13.19) a.re continuous on D for all fixed (e, 1]), and the function (13.20) is bounded
on D. This immediately implies that g(t,s,"') is of bounded varia.- tion and g(.",1',O') is weakly continuous. By Radon's theorem, the operator (13.2) is bounded on C(D). Finally, let us prove the formula (13.21) for the norm of the operator (13.2). We have (13.25) I/PII.c(O) = sup {IIPz(t, s)IIo-: (t, s) ED}, where pz(t, s) is considered, for fixed (t, s) E D, as a bounded linear functional on C(D). Therefore it suffices to prove that VarDg(t, s,',') = IIPz(t, s)110-for fixed (t, s) ED. Given an arbitrary functional f E C·, let
where
g(T,U) := o if
a < T Shand c < u S d, T = a or u = c,
{ 1 for a SuS T and c S v S U,
Z'f'tt(u, v) := • 0 for T < u S h or u < v S d.
