ABSTRACT
Various conditions foi the weak continuity. of vector functions are reported in GLIVENKO [1936]. The most appropriate one is the Riesz criterion which states that the vector function (2.8) is weakly continuous on BY if and only if (2.8) is bounded and the scalar functions s ~ g·(s)(b) - g·(s)(a) and s ~ f: g·(s)(O') do' are continuous. By standard formulas (see e.g. NATANSON [1950]), the total variation of g.(s) on [4, b] is
/Ig·(s)IIBv = Var[CI,bJ9·(S) = Ic(s)1 +1b Ik(s, 0')1 do' = 1(s); consequently, the boundedness of the vector function (2.8) in BY is equivalent to the boundedness of the function (2.4). Moreover, by the obvious formulas
1-g·(s)(O')dO' = l-[c(s)x(s, 0') +1" k(s,r)dr]dO' = P(z,s), the boundedness of the operator (2.1) on C implies the continuity of both functions (2.2) and (2.3). Thus we have shown that, whenever the operator (2.1) acts in the space C, the functions (2.2) and (2.3) are continuous, and the function (2.4) is bounded. The converse is also true: if the functions (2.2) and (2.3) are continuous and the function (2.4) is bounded, then the vector function (2.8) is weakly continuous on the space BV, and this is in turn equivalent to the fact that the operator (2.1) acts in the space C. It remains to prove the relation (2.5). But this is an immediate consequence of the formula
/I A /I.c<C) = sup sup (b z(0') dg·( s)(0')CI~.~b IIzllc~l lCl = sup IIg·(s)/IBv = sup 1(s).
