In this chapter we complete the proof of the Hypercontractivity Theorem for uniform $\pm 1$ bits. We then generalize the $(p,2)$ and $(2,q)$ statements to the setting of arbitrary product probability spaces, proving the following:

The General Hypercontractivity TheoremLet $(\Omega_1, \pi_1), \dots, (\Omega_n, \pi_n)$ be finite probability spaces, in each of which every outcome has probability at least $\lambda$. Let $f \in L^2(\Omega_1 \times \cdots \times \Omega_n, \pi_1 \otimes \cdots \otimes \pi_n)$. Then for any $q > 2$ and $0 \leq \rho \leq \frac{1}{\sqrt{q-1}} \cdot \lambda^{1/2-1/q}$, \[ \|\mathrm{T}_\rho f\|_q \leq \|f\|_2 \quad\text{and}\quad \|\mathrm{T}_\rho f\|_2 \leq \|f\|_{q'}. \] (In fact, the upper bound on $\rho$ can be slightly relaxed to the value stated in Theorem 17 of this chapter.)

We can thereby extend all the consequences of the basic Hypercontractivity Theorem for $f : \{-1,1\}^n \to {\mathbb R}$ to functions $f \in L^2(\Omega^n, \pi^{\otimes n})$, except with quantitatively worse parameters depending on “$\lambda$”. We also introduce the technique of randomization/symmetrization and show how it can sometimes eliminate this dependence on $\lambda$. For example, it’s used to prove Bourgain’s Sharp Threshold Theorem, a characterization of boolean-valued $f \in L^2(\Omega^n, \pi^{\otimes n})$ with low total influence which has no dependence at all on $\pi$.

Is $q’$ defined here?