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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## Recent comments

Gautam Kamath: Is $q'$ defined here?Gautam Kamath: On this page, Hölder is displaying for me as H{ö}lder - is t...Ryan O'Donnell: Yes, you're right. This is not a well-written proof by the ...Ryan O'Donnell: Thanks, yes.Ryan O'Donnell: Yep, thanks.Ryan O'Donnell: Yep, thanks.Noam Lifshitz: I think that in exercise 23, it should be $(p,2)$ Hypercontr...