Multilinear Duality and Factorisation for BrascampLiebtype Inequalities with applications
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 MaureyNikisinStein 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, functionalanalytic considerations concerning the dual of $L^\infty$, and the YosidaHewitt 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 LoomisWhitney inequalities, BrascampLieb inequalities and multilinear Kakeya inequalities.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1809.02449
 Bibcode:
 2018arXiv180902449C
 Keywords:

 Mathematics  Functional Analysis;
 46A20;
 46B10;
 46G25;
 47H60;
 47N99;
 49J35;
 42Bxx
 EPrint:
 Title shortened. Accepted for publication in Jour. Eur. Math. Soc. Paper streamlined and restructured. Several results now have more succinct and general statements and proofs. References added