278 Boolean Analysis — Extension
278.1 Overview
Reverse Bonami–Beckner hypercontractivity extends to finite real exponents on both sides of zero, including the geometric-mean and exponent-one cases.
278.2 Declarations
Let \(q\le p\le 1\), \(q\lt 1\), \(0\le \rho \le 1\), and \(\rho ^2\le (1-p)/(1-q)\). Every \(f:\{ 0,1\} ^n\to [0,\infty )\) satisfies \(M_q(T_\rho f)\ge M_p(f)\). For nonnegative functions \(u\), the extended mean is \(M_r(u)=(\mathbb {E}u^r)^{1/r}\) when \(r\gt 0\); the same formula applies for \(r\lt 0\) when \(u\) is positive everywhere. For positive \(u\), \(M_0(u)=\exp (\mathbb {E}\log u)\); if \(r\le 0\) and \(u\) has a zero, \(M_r(u)=0\). All expectations are uniform. 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 \(q\le p\le 1\), \(q\lt 1\), \(0\le \rho \le 1\), and \(\rho ^2\le (1-p)/(1-q)\). Every \(f:\{ 0,1\} ^n\to [0,\infty )\) satisfies \(M_q(T_\rho f)\ge M_p(f)\). For nonnegative functions \(u\), the extended mean is \(M_r(u)=(\mathbb {E}u^r)^{1/r}\) when \(r\gt 0\); the same formula applies for \(r\lt 0\) when \(u\) is positive everywhere. For positive \(u\), \(M_0(u)=\exp (\mathbb {E}\log u)\); if \(r\le 0\) and \(u\) has a zero, \(M_r(u)=0\). All expectations are uniform. 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.