2015/06/30 by O. Delgado, Delgado, O., E. A. Sánchez Pérez +1
Mathematics · #46E30 #47B38 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46E30 #msc:47B38
paper · pdf · doi:10.48550/arxiv.1506.09010
arxiv created 2015/06/30 · arxiv updated 2015/07/01
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 Lp(μ) and Lq(ξ) for constructing the new space SXp 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 SXp q(ξ). This result turns out to be a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provide (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, opening the door to the study of the multilinear versions of q-summing operators also in these cases.