Let’s now study hypercontractivity for general random variables. By the end of this section we will have proved the General Hypercontractivity Theorem stated at the beginning of the chapter.
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Let’s now study hypercontractivity for general random variables. By the end of this section we will have proved the General Hypercontractivity Theorem stated at the beginning of the chapter. [...] In this section we’ll prove the full Hypercontractivity Theorem for uniform $\pm 1$ bits stated at the beginning of Chapter 9: [...] 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 Theorem Let $(\Omega_1, \pi_1), \dots, (\Omega_n, \pi_n)$ be finite probability spaces, in each of which [...] The history of the Hypercontractivity Theorem is complicated. [...] [...] Recalling the social choice setting of Chapter 2.5, consider a $2$candidate, $n$voter election using a monotone voting rule $f : \{1,1\}^n \to \{1,1\}$. We assume the impartial culture assumption (that the votes are independent and uniformly random), but with a twist: one of the candidates, say $b \in \{1,1\}$, is able to secretly bribe $k$ [...] With the $(2,q)$ and $(p,2)$Hypercontractivity Theorems in hand, let’s revisit some applications we saw in Sections 1 and 2. [...] At this point we have established that if $f : \{1,1\} \to {\mathbb R}$ then for any $p \leq 2 \leq q$, \[ \\mathrm{T}_{\sqrt{p1}} f\_2 \leq \f\_p, \qquad \\mathrm{T}_{1/\sqrt{q1}} f\_q \leq \f\_2. \] We would like to extend these facts to the case of general $f : \{1,1\}^n \to {\mathbb R}$; i.e., establish the $(p,2)$ [...] Although you can get a lot of mileage out of studying the $4$norm of random variables, it’s also natural to consider other norms. [...] An immediate consequence of the Bonami Lemma is that for any $f : \{1,1\}^n \to {\mathbb R}$ and $k \in {\mathbb N}$, \begin{equation} \label{eqn:24hyperconk} \\mathrm{T}_{1/\sqrt{3}} f^{=k}\_4 = \tfrac{1}{\sqrt{3}^k} \f^{=k}\_4 \leq \f^{=k}\_2. \end{equation} [...] 

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