2018/09/07 by Anthony Carbery, Carbery, Anthony, Timo S. Hänninen +3 · 1 citation
Mathematics · #42Bxx #46A20 #46B10 #46G25 #47H60 #47N99 #49J35 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1809.02449
openalex publication_date 2018/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L1. The development of the theory involves convex optimisation and minimax theory, functional-analytic considerations concerning the dual of L^∞, 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.