ABSTRACT
Theorem 12.3. Let Ul and U'l be two ideal spaces over T, and Vl and V'l two ideal spaces over S. Suppose that the linear integral operator (12.17) maps Ul into U'l' Then the partial integral operator (12.4) acts between the spaces X = [Ul 4-Vl] and Y = [U'l 4-V'l ], is regular, and satisfies
= II(t, r) 1-+ IIl(t,', r)lIv2fVsIl3Ws,u2)' Similarly, if the linear integral operator (12.18) maps Vl into V'l , then the partial integral operator (12.5) acts between the spaces X :;:: [Ul ~ Vl] and Y =[U'l ~ V'l] is regular, and satisfies
= II{s,O') 1-+ IIm(.,s,0')IIu2/usI13(Vs'V2)'
o We prove a.gain only the first statement. Obviously, for x EX, we have
IILx{t, ·)IIv2 = 11£ let, ',r)x{r, .)dP{r)llv2 ~£i{t, r)lIx{r, .)lIvsdp{r) = Lu{t),
where u(t) = IIx(t, .)lIvs' Consequently,
We conclude this subsection with two more acting conditions for partial integral operators in spaces with mixed norm; these conditions build on multiplicator spaces of functions of two variables.
