IAIIBI

AB E .cb(Lp ). Moreover, the evident inequality IABI $ and the definition (3.10) of the norm in .cb(Lp ) imply that

and IIIAI II.c(L,.) II IBIII.c(L,.) = IIAII.ct(L,.) IIBII.cuL,.) ,

and thus the first assertion is proved. The second assertion follows immediately from (3.15): in fact, if A, say, is a "pure" integral operator, then c(s) == 0, hence c(s)d(s) == 0, and thus AB is also an integral operator.•

3.3. Sufficient conditions for regularity In this subsection we shall formulate some important sufficient conditions for the operator (3.1) to belong to the class .cHLp). Such conditions may be obtained, loosely speaking, from sufficient conditions for the integral operator (3.9) to be regular in the space Lp (KRASNOSEL'SKIJ-ZABREJKO-PUSTYL'NIK-SOBOLEVSKIJ [1966], ZABREJKO-KoSHELEV-KRASNOSEL'SKIJ-MIKHLIN-RAKOVSHCHIKSTETSENKO [1968]). For our purpose, however, it is more convenient to formulate such conditions through imbedding theorems for classes of kernels on [a, b] x [a, b]. Following KRASNOSEL'SKIJ-ZABREJ KOPUSTYL'NIK-SOBOLEVSKIJ [1966], we denote by 3p (1 $ p $ (0) the (Zaanen) class of all measurable functions z = z(s, 0') on [a,b] x [a, b) for which the norm

(q = pj(p -1» is finite. The importance of this class is seen from the fact that every function in 3p is the kernel of some regular linear integral operator in Lp , and, conversely, every kernel of a regular linear integral operator in Lp is a function in 3p • Combining this fact with Lemma. 3.2 we a.rrive at the following important

We can reformulate this result as follows. IT we denote, in analogy to Subsection 2.3, by £m(Lp ) the class of all multiplication operators (3.8) in Lp (note that a multiplication operator in Lp is always regular), and by .cHLp ) the class of all regular integral operators (3.9) in Lp , then the decomposition

holds. In particular, we cannot find a parallel counterexample to Example 2.1 for regular Barbashin operators in Lebesgue spaces. At first glance it seems that the problem of describing the class £t(Lp ) is completely solved by Lemma 3.6. However, there is an essential Haw: it is very hard, in general, to find explicit formulas or estimates for the norm of a function in 3p , and just the problem of characterizing the elements in 3p is not solved even for very simple functions arising frequently in applications (e.g. functions with polynomial or logarithmic singularities on the diagonal 8 = 0'). Conditions which are merely sufficient, of course, are well known (see KRASNOSEL'- SKIJ-ZABREJKO-PUSTYL'NIK-SOBOLEVSKIJ [1966] and below). Some more auxiliary classes of kernels on [a, b) x [a, b] are in order. Given a measurable set D ~ [a, b], let

(3.19) p. () ._ {Z(S) if SED,DZ s.- 0 if s ¢ D

denote the characteristic operator of D ~ [a, b], i.e. the multiplication operator generated by the characteristic function Xv of D. By 3; we denote the space of all z E 3p such that

(3.20) lim sup 1616IZ(s,q)Pvz(q)y(s)1 dcTds = O. #(V)-O IIsllLp Sl G G

Similarly, by 3; we denote the space of all z E 3p such that

It is well known that in case 1 < p < 00 all three classes 3;,3; and 3~ coincide and consist of all kernels which generate compact regular integral operators (3.9) in Lp (more precisely, they generate all regular operators (3.9) such that both K and IKI are compact). In case p = 1 we have 31 = {O}, and 3t consists of all kernel functions which generate weakly compact integral operators in Ll. Similarly, in case p = 00 we have 3t, ={O}, and 3;" consists of all kernels which generate weakly compact integral operators in Loo • As already mentioned, the kernels k E 3p are precisely those which generate integral operators K E .ci(Lp). Let us denote by .c~'-(Lp), .ci'+(Lp), and .ci,o(Lp) the class of all integral operators K E .ci(Lp) which are generated by kernels k E 3; ,3;, and 3~, respectively. From well known properties of compact and weakly compact operators we get the following useful result (which, by the way, may also be obtained directly):

Now we introduce some important subclasses of the kernel class 3p • For 1 ~ p, q ~ 00, denote by U(p, q) the linear space of all measurable

70 Chapter I: Equations of Barbaahin Type

functions z = z(8,0') on [a, b] x [a, b] for which the so-called mixed norm

(3.22) is finite; similarly, denote by yep, q) the linear space of all measurable functions z =Z(8, 0') on [a, b] x [a, Ii] for which the mixed norm

(3.23) is finite. The functionals (3.22) and (3.23) are norms which turn U(p, q) and yep, q) into Banach spaces. The following two propositions provide some relations between the kernel classes introduced so far; they may be regarded as reformulation of well known sufficient acting conditions for integral operators in Lp , expressed in terms of the corresponding kernels (HILLETAMARKIN [1930], see also KANTOROVICH-AKILOV [1977] or KRASNOSEL'SKIJ-ZABREJKO-PUSTYL'NIK-SOBOLEVSKIJ [1966]): Lem.ma 3.8. Let 1 ~ p ~ 00 and q = pl(p - 1). Then U(q,p) is continuously imbedded in 3p , i.e.