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 Lq (ξ ) for constructing the new space S q X p (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 q X p (ξ ). Our arguments lead to a mixture of the Pietsch and MaureyRosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through Lq -spaces when 1 ≤ q ≤ p. Thus, our result completes the picture, showing what happens in the complementary case 1 ≤ p ≤ q.