Maurey-Rosenthal domination for abstract Banach lattices

Abstract

We extend the Maurey-Rosenthal theorem on integral domination and factorization of p-concave operators from a p-convex Banach function space through [InlineEquation not available: see fulltext.]-spaces for the case of operators on abstract p-convex Banach lattices satisfying some essential lattice requirements-mainly order density of its order continuous part-that are shown to be necessary. We prove that these geometric properties can be characterized by means of an integral inequality giving a domination of the pointwise evaluation of the operator for a suitable weight also in the case of abstract Banach lattices. We obtain in this way what in a sense can be considered the most general factorization theorem of operators through [InlineEquation not available: see fulltext.]-spaces. In order to do this, we prove a new representation theorem for abstract p-convex Banach lattices with the Fatou property as spaces of p-integrable functions with respect to a vector measure. MSC: 46G10, 46E30, 46B42. © 2013 Juan and Sánchez Pérez; licensee Springer.