Multilinear duality and factorisation for Brascamp–Lieb-type inequalities

  • Anthony Carbery

    University of Edinburgh, UK
  • Timo S. Hänninen

    University of Helsinki, Finland; University of Edinburgh, UK
  • Stefán Ingi Valdimarsson

    Arion Bank, Reykjavík; University of Iceland, Reykjavík, Iceland
Multilinear duality and factorisation for Brascamp–Lieb-type inequalities cover
Download PDF

This article is published open access under our Subscribe to Open model.

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–Nikishin–Stein theory of factorisation of operators, and present a fully multilinear version of Maurey’s fundamental theorem on factorisation of operators through . The development of the theory involves convex optimisation and minimax theory, functional-analytic considerations concerning the dual of , and the Yosida–Hewitt theory of finitely additive measures. We consider 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.

Cite this article

Anthony Carbery, Timo S. Hänninen, Stefán Ingi Valdimarsson, Multilinear duality and factorisation for Brascamp–Lieb-type inequalities. J. Eur. Math. Soc. 25 (2023), no. 6, pp. 2057–2125

DOI 10.4171/JEMS/1229