Strong extensions for q-summing operators acting in p-convex Banach function spaces for 1≤p≤q

Let 1 ≤ p ≤ q < ∞ and let X be a p -convex Banach function space over a σ -finite measure μ . We combine the structure of the spaces L p ( μ ) and L q ( ξ ) for constructing the new space S X p q ( ξ ) , where ξ is a probability Radon measure on a certain compact set associated to X . We show some of its properties, and the relevant fact that every q -summing operator T defined on X can be continuously (strongly) extended to S X p q ( ξ ) . Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q -summing operators through L q -spaces when 1 ≤ q ≤ p . Thus, our result completes the picture, showing what happens in the complementary case 1 ≤ p ≤ q .