CS 880 : Advanced Complexity Theory 2 / 15 / 2008 Lecture 10 : Hypercontractivity

the inequality follows from Hölder’s inequality: E [ fg] ≤ ∥f ∥∥ p ∥g ∥∥ q , if 1p + 1 q = 1 with p, q ≥ 1. If α = ±1 then (1) fails unless p = q or f is constant in absolute value. This follows because (T±1f)(x) = f(±x), where −x denotes x with all its bits flipped, and because the only functions f for which ∥f ∥∥ p = ∥f ∥∥ q for p 6= q are those that are constant in absolute value. In proving the theorem we will arrive at the condition in the statement of the theorem on |α| as the weakest one that guarantees (1) to hold.