Analysis of Boolean Functions ( CMU 18 - 859 S , Spring 2007 ) Lecture 16 : The Hypercontractivity Theorem

1 Statement and history In this lecture, we prove the full-blown hypercontractivity theorem for {−1, 1} n. The idea behind the statement is that the T ρ operator smooths, or " reasonable-izes " , functions. Some examples, including two that were mentioned in the last lecture: Example 1.2 • q = 4, p = 2, ρ = 1/ √ 3: T 1/ √ 3 f 4 ≤ f 2 • q = 2, p = 4/3, ρ = 1/ √ 3: T 1/ √ 3 f 2 ≤ f 4/3 • q = q, p = 2, ρ = 1/ √ q − 1: T 1/ √ q−1 f 4 ≤ f 2 One corollary of the last of these is often quite sufficient; it's also a generalization of the original (2, 4, 1/ √ 3)-hypercontractivity result we proved easily by induction: Proof: f 2 q = d k=0 f =k 2 q = T 1/ √ q−1 d k=0 q − 1 k f =k 2 q ≤ d k=0 q − 1 k f =k 2 2 = d k=0 (q − 1) k |S|=kˆf (S) 2 ≤ (q − 1) d S ˆ f (S) 2 = (q − 1) d f 2 2 ,