DAN

Abstract

A hyper-contractive inequality for an operator T states that |Tf|_q \leq |f|_p where q > p > 1 for all functions f. Hyper contractive inequalities play a crucial role in analysis in general and in discrete Fourier analysis in particular.

A reverse hyper-contractive inequality for the operator T states that |Tf|_q \geq |f|_p for q < p < 1 (q and p can be negative) and all strictly positive functions f.

The first reverse hyper-contractive inequalities were proved by Borell more than 2 decades ago. While these inequalities may look obscure, they have been used for the solution of a number of problems in the last decade. I will survey applications of the inequalities and discuss new results relating reverse hyper-contractive inequalities to hyper-contractive, Log-Sobolev and Poincare inequalities as well as some new applications.

This is a joint work with K Oleszkiewicz (Warsaw) and A Sen (Cambridge).