[...]


[...] In this section we’ll prove the full Hypercontractivity Theorem for uniform $\pm 1$ bits stated at the beginning of Chapter 9: [...] [...] At this point we have established that if $f : \{1,1\} \to {\mathbb R}$ then for any $p \leq 2 \leq q$, \[ \\mathrm{T}_{\sqrt{p1}} f\_2 \leq \f\_p, \qquad \\mathrm{T}_{1/\sqrt{q1}} f\_q \leq \f\_2. \] We would like to extend these facts to the case of general $f : \{1,1\}^n \to {\mathbb R}$; i.e., establish the $(p,2)$ [...] 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 $0 \leq \rho \leq \sqrt{\tfrac{p1}{q1}}$. [...] 

Copyright © 2016 Ryan O'Donnell  All Rights Reserved 
Recent comments