[...]


[...] [...] In Section 3 we collected a number of consequences of the General Hypercontractivity Theorem for functions $f \in L^2(\Omega^n, \pi^{\otimes n})$. All of these had a dependence on “$\lambda$”, the least probability of an outcome under $\pi$. This can sometimes be quite expensive; for example, the KKL Theorem and its consequence Theorem 28 are trivialized [...] 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. [...] [...] 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. [...] 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 Fourier expansion for realvalued boolean functions dates back to Walsh [Wal23] who introduced a complete orthonormal basis for $L^2([0,1])$ consisting of $\pm 1$valued functions, constant on dyadic intervals. [...] 

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