MULTILINEAR DUALITY AND FACTORISATION FOR BRASCAMP--LIEB-TYPE INEQUALITIES WITH APPLICATIONS

Anthony Carbery, Timo Hanninen, stefan valdimarsson

Research output: Contribution to journalArticlepeer-review

Abstract

We initiate the study of a duality theory which applies to norm inequalities for
pointwise weighted geometric means of positive operators. The theory finds its expression in terms of certain pointwise factorisation properties of function spaces which are naturally associated to the norm inequality under consideration. We relate our theory to the Maurey--Nikisin--Stein theory of factorisation of operators, and present a fully multilinear version of
Maurey's fundamental theorem on factorisation of operators through L^1. The development of the theory involves convex optimisation and minimax theory, functional-analytic considerations concerning the dual of L^\infty, and the Yosida--Hewitt theory of finitely additive measures.
We explore the connections of the theory with the theory of interpolation of operators. We discuss the ramifications of the theory in the context of concrete families of geometric inequalities, including Loomis--Whitney inequalities, Brascamp--Lieb inequalities and multilinear Kakeya inequalities.
Original languageEnglish
JournalJournal of the European Mathematical Society
Publication statusAccepted/In press - 26 Sep 2020

Keywords

  • 42B-99 Harmonic Analysis in Several Variables
  • 47H560 Multilinear and polynomial operators

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