The history of the Hypercontractivity Theorem is complicated.
[...]


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} [...] As anyone who has worked in probability knows, a random variable can sometimes behave in rather “unreasonable” ways. It may be never close to its expectation. It might exceed its expectation almost always, or almost never. It might have finite $1$st, $2$nd, and $3$rd moments, but an infinite $4$th moment. All of this poor behaviour [...] In 1970, Bonami proved the following central result: The Hypercontractivity Theorem Let $f : \{1,1\}^n \to {\mathbb R}$ and let ${1 \leq p \leq q \leq \infty}$. Then $\\mathrm{T}_\rho f\_q \leq \f\_p$ for $0 \leq \rho \leq \sqrt{\tfrac{p1}{q1}}$. [...] The origins of the orthogonal decomposition described in Section 3 date back to the work of Hoeffding [Hoe48] (see also [vMis47]). [...] 

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