-. -.

Similarly, if the function (t, s) 1-+ lImet,S, .)lIv; belongs to the multiplicator space [LI - Un/[U~ -. V;], then the partial integral operator (12.5) acts between the spaces X = (UI - VI] and Y = (U2 -. V2] is regular and satisfies

o The proof is completely analogous to that of Theorem 12.4; one has just to pass to the correspon'ding associate operators. •

All sufficient conditions of Theorems 12,1 - 12.5 are different for the partial integral operator (12.4) and the partial integral operator (12.5) and refer to different kernel spaces with mixed norm. In this wa.y, the above theorems contain eight statements which guarantee the acting (and, except for Theorem 12.1, also the regularity) of partial integral operators between four possible combinations of spaces with mixed norm,

12.2. Lebesgue spaces with mixed norm In Subsection 11.5 we had to show that certain functions of two variables, constructed by means of the kernels I = I(t, 8, r) and m = m(t, s, 0'), belong to certain ideal spaces, constructed by means of the spaces UI, U2, l't and V2' Since these ideal spaces are rather complicated, however, the natural. problem arises to replace them by simpler and more tractable ones, One possibility to do so is to introduce ideal spaces of functions defined either on T x S X T, or on T x S X S, or on some permutation of these Cartesian products. Denote by () = «()1' ()2, ()3) an arbitra.ry permuta.tion of the arguments (t,8, r) E T x S X T, or (t,s, 0') E T x S x S. Given three ideal spaces

W h W2, and W3, by [Wh W2, W3i 8] we denote the ideal space of all functions w of three variables for which the norm

is defined and finite. Using classical results on linear integral operators, from Theorems 12.2 - 12.5 we get the following

Theorem 12.6. Let Ut and U2 be two ideal spaces over T, and Vt and V2 two ideal spaces over S. Suppose that I E [U2, V2/Vt, UIj I] for some 1 = (Ih 12, ( 3 ), Then the partial integral operator (12.4) acts between X and Y, is regular, and satisfies (12.25) Here the spaces X and Y have to be chosen according to the formula

X =[Ut - VI], Y = [U2 - V2] if 9 =(S,t,T) or 8 =(S,T,t) X = [Ut - Vt], Y = [U2 - V2] if 1 = (t,T,S) or 8 =(T,t,S), X = [Ut - Vt], Y = [U2 - V2] if 8 = (t,S,T), X = [Ut - VIl, Y = [U2 - V2] if 8 = (T,S,t).