Chapter 10 notes

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Chapter 10 exercises

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§10.4: More on randomization/symmetrization

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 [...]

§10.2: Hypercontractivity of general random variables

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.

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Chapter 9 exercises

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§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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Chapter 9: Basics of hypercontractivity

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 $0 \leq \rho \leq \sqrt{\tfrac{p-1}{q-1}}$.

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Chapter 1 notes

The Fourier expansion for real-valued 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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