In this section we will collect some applications of the General Hypercontractivity Theorem, including generalizations of the facts from Section 9.5.
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In this section we will collect some applications of the General Hypercontractivity Theorem, including generalizations of the facts from Section 9.5. [...] With the $(2,q)$ and $(p,2)$Hypercontractivity Theorems in hand, let’s revisit some applications we saw in Sections 1 and 2. [...] 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} [...] The most obvious spectral property of a truly random function $\boldsymbol{f} : \{1,1\}^n \to \{1,1\}$ is that all of its Fourier coefficients are very small (as we saw in Exercise 5.8). [...] 

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