ABSTRACT
Apart from the acting and regularity conditions for the partial integral operators (12.4) and (12.5), many other statements may be formulated. For instance, one may obtain further statements by means of interpolation theory (applied to either classical or partial integral operators, see KALITVIN-MILOVIDOV [1981] and POVOLOTSKIJ-KALITVIN [1983, 1985]), of Kantorovich type theorems, or of other methods. We point out that Theorem 12.1 and its partial converse considered above imply the following useful criterion: the partial integral operator (12.4) (respectively, (12.5» acts in Lp if and only if all operators of the family (12.6) (respectively, (12.7» act in L'P for each s E S (respectively, t E T) and have uniformly bounded norms. The statements of Theorems 12.7 and 12.8, referring to the Lebesgue type spaces [Lp - L9 ] and [L'P +- L9], carryover as well to the Orlicz type spaces [LM - LN] and [LM +- LN], as we shall show in the following subsection. The following theorem which is taken from KALITVIN [1995a] gives an effective acting criterion for the operator (11.1) in Loo =Loo(T x S) and in L1 = L1(T x S).
