In  Cameron and Storvick introduced a very general operator-valued function space “integral”. In [3-5, 8, 9, 11, 13-20] the existence of this integral as an operator from L2
to L2 was established for certain functions. Recently the existence of the integral as an operator from L1 to L∞, has been studied [6, 7, 21]. In this paper we study the integral as an operator from Lp
to Lp′, where 1 < p ≤ 2. The resulting theorems extend the theory substantially and indicate relationships between the L2-L2
theories that were not apparent earlier. Even in the most studied case, p = p′ = 2, the results below strengthen the theory.