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.
Continue reading §9.3: $(2,q)$ and $(p,2)$hypercontractivity for a single bit


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 can cause a lot of trouble — wouldn’t it be nice to have a class of “reasonable” random variables? In 1970, Bonami proved the following central result:
Continue reading Chapter 9: Basics of hypercontractivity The origins of the orthogonal decomposition described in Section 3 date back to the work of Hoeffding [Hoe48] (see also [vMis47]). A decision tree $T$ for $f : \{1,1\}^n \to \{1,1\}$ can be thought of as a deterministic algorithm which, given adaptive query access to the bits of an unknown string $x \in \{1,1\}^n$, outputs $f(x)$. E.g., to describe a natural decision tree for $f = \mathrm{Maj}_3$ in words: “Query $x_1$, then $x_2$. If they are equal, output their value; otherwise, query and output $x_3$.” For a worstcase input (one where $x_1 \neq x_2$) this algorithm has a cost of $3$, meaning it makes $3$ queries. The cost of the worstcase input is the depth of the decision tree. The previous section covered the case of $f \in L^2(\Omega^n, \pi^{\otimes n})$ with $\Omega = 2$; there, we saw it could be helpful to look at explicit Fourier bases. When $\Omega \geq 3$ this is often not helpful, especially if the only “operation” on the domain is equality. For example, if $f : \{\mathsf{Red}, \mathsf{Green}, \mathsf{Blue}\}^n \to {\mathbb R}$ then it’s best to just work abstractly with the orthogonal decomposition. However if there is a notion of, say, “addition” in $\Omega$ then there is a natural, canonical Fourier basis for $L^2(\Omega, \pi)$ when $\pi$ is the uniform distribution. Perhaps the most common generalized domain in analysis of boolean functions is the case of the hypercube with “biased” bits. 

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