Chapter 10 notes

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§10.1: The Hypercontractivity Theorem for uniform $\pm 1$ bits

In this section we’ll prove the full Hypercontractivity Theorem for uniform $\pm 1$ bits stated at the beginning of Chapter 9:

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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 $(p,2)$- [...]

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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