Integration and Linear Operations

1. Notation and introduction. By LP (1 ^ p < oo ) will be understood the class of real valued measurable functions c/>(P), O^P^l, for which f \iP) \pdP< oo, by SPiX) the class of measurablef functions/(P) on (0,1) to the space X of type B [seej 1, p. 53] for which /J|/(P)||pdP< oo. With ||/||= {/J|/(P)||pdP}1/",5P(Ar) is a Banach space. In case p > 1, Lp. is defined by the equality p,=1p/ip—i) and if p = 1 the symbol LP> = L„ = M will stand for the space of real, essentially bounded and measurable functions with ||c/>|| = ess. sup. |c/>(P)|. Similarly for Sp-iX) and S«,(X), and for brevity we write Spq in place of SPiLq). By Lpq we mean the class of real valued measurable functions KiP, Q) that belong to 7, for each P and for which [f0\KiP, Q)\qdQ]il" belongs to Lp. Finally the term linear operation is used in the sense of Banach, i.e., for an additive continuous function. In terms of this notation our results are described in the following paragraph. It is easily shown that a kernel KiP, Q) in LP'q defines a linear operation