Finite bounds for Holder-Brascamp-Lieb multilinear inequalities
  A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalities of Holder, Young, and Loomis-Whitney. This is a companion to a recent paper by the same authors, containing a different proof and discussing certain variants. 
Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao