A variational proof of a disentanglement theorem for multilinear norm inequalities

TL;DR

This paper provides a simple, variational proof of a fundamental disentanglement theorem for multilinear norm inequalities, linking estimates on geometric means to separate family estimates, with implications for classical and duality results.

Contribution

It introduces an elementary, variational proof of the disentanglement theorem, simplifying previous complex methods involving minimax and measure theory.

Findings

Simplified proof of the disentanglement theorem

Connections to classical factorisation theorems

Foundation for duality theory in multilinear inequalities

Abstract

The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately. On the one hand, the theorem gives a uniform approach to classical results including Maurey's factorisation theorem and Lozanovski\u{\i}'s factorisation theorem, and, on the other hand, it underpins the duality theory for multilinear norm inequalities developed in our previous two papers. In this paper we give a simple proof of this basic disentanglement theorem. Whereas the approach of our previous paper was rather involved - it relied on the use of minimax theory together with weak*-compactness arguments in the space of finitely additive measures, and an application of the Yosida-Hewitt theory of such measures - the alternate approach of this paper is rather straightforward: it instead depends upon elementary perturbation and compactness arguments.