Some Applications of Global Hypercontractivity in Matrix Groups

The classical hypercontractive inequality for the noise operator on the discrete cube plays a central role in the Analysis of Boolean functions. For other domains 'beyond the cube', analogous hypercontractivity results hold provided that the functions are sufficiently pseudorandom, i.e., global. Using improved global hypercontractive inequalities of Evra, Kindler, and Lifshitz on the space of bilinear forms, we give new Erdős--Sós-type results for various matrix spaces and prove a polynomial Bogolyubov-type lemma for finite groups of Lie type. This is joint work with Shai Evra, Esty Kelman, Guy Kindler, Noam Lifshitz, and Ohad Sheinfeld.