§9.4: Two-function hypercontractivity and induction

At this point we have established that if $f : \{-1,1\} \to {\mathbb R}$ then for any $p \leq 2 \leq q$, $\|\mathrm{T}_{\sqrt{p-1}} f\|_2 \leq \|f\|_p, \qquad \|\mathrm{T}_{1/\sqrt{q-1}} f\|_q \leq \|f\|_2.$ We would like to extend these facts to the case of general $f : \{-1,1\}^n \to {\mathbb R}$; i.e., establish the $(p,2)$- [...]

§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.

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§9.2: Small subsets of the hypercube are noise-sensitive

An immediate consequence of the Bonami Lemma is that for any $f : \{-1,1\}^n \to {\mathbb R}$ and $k \in {\mathbb N}$, $$\label{eqn:2-4-hypercon-k} \|\mathrm{T}_{1/\sqrt{3}} f^{=k}\|_4 = \tfrac{1}{\sqrt{3}^k} \|f^{=k}\|_4 \leq \|f^{=k}\|_2.$$

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§9.1: Low-degree polynomials are reasonable

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 [...]

§8.6: Highlight: Randomized decision tree complexity

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 [...]

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§8.1: Fourier bases for product spaces

We will now begin to discuss functions on (finite) product probability spaces.

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