Multilinear duality and factorisation for Brascamp–Lieb-type inequalities

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