Fix a dimension \(n\) and real exponents \(p, u\) with \(1 \lt p \le u \le 2\), and set the noise rate

Then for every function \(f : \{ 0,1\} ^n \to \mathbb {R}\) on the Boolean hypercube, the noise operator \(T_\rho \) contracts \(L^p\) into \(L^u\):

where the expectations 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.

Fix a natural number \(n\). Let \(f : \{ 0,1\} ^n \to \mathbb {R}\) be a Boolean function, and let \(p, u\) be real exponents with \(2 \le p \le u\). Suppose \(\rho \in \mathbb {R}\) satisfies \(0 \le \rho \le 1\) together with \(\rho \le \sqrt{(p-1)/(u-1)}\). Then, with expectations taken under the uniform measure on \(\{ 0,1\} ^n\) and \(T_\rho \) the noise operator, the higher \(L^u\) norm of the noised function is bounded by the lower \(L^p\) norm of \(f\):

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 \(n\). Let \(1 \le p \le u\) with \(u \gt 1\), and let \(\rho \) satisfy \(0 \le \rho \le 1\) together with \(\rho \le \sqrt{(p-1)/(u-1)}\). Then for every Boolean function \(f:\{ 0,1\} ^n \to \mathbb {R}\), the noise operator \(T_\rho \) contracts the \(L^u\) norm below the \(L^p\) norm:

where the expectations are taken over 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.

Fix a natural number \(n\), real exponents \(1 \le p \le u\) with \(u \ge 2\), and a noise rate \(\rho \) with \(0 \le \rho \le 1\) satisfying \(\rho \le \sqrt{(p-1)/(u-1)}\). Then for every pair of Boolean functions \(f, g : \{ 0,1\} ^n \to \mathbb {R}\),

where \(\langle \cdot ,\cdot \rangle \) is the \(L^2\) inner product, \(\mathbb {E}\) the expectation, and \(T_\rho \) the noise operator, all taken with respect to the uniform measure on the hypercube. 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.