Let \(n\) be a natural number, let \(0\le \rho \le 1\), and fix \(x\in \{ 0,1\} ^n\). The noise kernel \(K_\rho (x,y)=\prod _{i=1}^{n}\frac{1+\rho \, \sigma (x_i)\, \sigma (y_i)}{2}\), where \(\sigma (0)=1\) and \(\sigma (1)=-1\), satisfies

Let \(n\) be a natural number, let \(0\le \rho \le 1\), and fix \(y\in \{ 0,1\} ^n\). The noise kernel \(K_\rho (x,y)=\prod _{i=1}^{n}\frac{1+\rho \, \sigma (x_i)\, \sigma (y_i)}{2}\), where \(\sigma (0)=1\) and \(\sigma (1)=-1\), satisfies

Thus \(K_\rho (\cdot ,y)\) is a probability distribution on the cube.

Let \(n\) be a natural number, and let \(T_\rho \) denote the noise operator with parameter \(\rho \in [0,1]\) acting on Boolean functions \(f : \{ 0,1\} ^n \to \mathbb {R}\). Then for every real exponent \(s \ge 1\) and every point \(x \in \{ 0,1\} ^n\),

where \(K_\rho (x,y) = \prod _{i} \tfrac {1 + \rho \, \sigma (x_i)\, \sigma (y_i)}{2}\) is the noise kernel, with \(\sigma (0)=1\) and \(\sigma (1)=-1\). For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(n\) be a natural number, let \(s \ge 1\) and \(\rho \in [0,1]\), and let \(f : \{ 0,1\} ^n \to \mathbb {R}\) be a Boolean function. Then, with expectations taken under the uniform measure on the cube, the noise operator \(T_\rho \) satisfies

For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Fix a natural number \(n\), exponents \(p,q\gt 1\) and a noise parameter \(\rho \) with \(0 \le \rho \le 1\), and write \(p' = p/(p-1)\) and \(q' = q/(q-1)\) for the Hölder conjugates. Then the noise operator \(T_\rho \) on Boolean functions \(f : \{ 0,1\} ^n \to \mathbb {R}\) satisfies the \((p \to q)\) bound

if and only if it satisfies the conjugate \((q' \to p')\) bound

where all norms are taken with respect to the uniform measure on \(\{ 0,1\} ^n\). For \(d\in \mathbb N\), \(h:\{ 0,1\} ^d\to \mathbb {R}\), and \(\tau \in \mathbb {R}\), the spectral noise operator is \(T_\tau h(x)=\sum _{S\subseteq [d]}\tau ^{|S|}\widehat h(S)\chi _S(x)\), where \(\chi _S(x)=(-1)^{\sum _{i\in S}x_i}\) and \(\widehat h(S)=\mathbb {E}[h\chi _S]\) under the uniform measure.

Let \(a\) and \(b\) be real numbers with \(0 \le a\), \(0 \lt b\), and \(a \le b\). Then \(\sqrt{a/b} \le 1\).

Let \(p, u \in \mathbb {R}\) with \(u \gt 1\). Then

Let \(p, u, \rho \) be real numbers with \(1 \lt p \lt u \lt 2\), and suppose \(\rho ^2 = (p-1)/(u-1)\). Then there exist real numbers \(\theta \in (0,1)\) and \(s \gt 0\) such that