293 Boolean Analysis — Basic
293.1 Overview
Affine hypercontractivity is defined for real random variables and extended norm exponents. Symmetric and Rademacher laws, finite atom support, multilinear evaluation, and moment identities provide the common probability-space framework.
293.2 Declarations
For a real function \(X\) on a measure space \((\Omega ,\mu )\) and extended exponents \(p,q\in [0,\infty ]\), \((p,q,\rho )\)-hypercontractivity means \(1\le p\le q\), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for every \(a,b\in \mathbb {R}\), with extended-valued norms and the essential supremum at infinity.
For a real function \(X\) on a measure space \((\Omega ,\mu )\) and \(q\in \mathbb {R}\), define \(N_q(X)\) by taking the extended norm at exponent \(\max (q,0)\) and converting it to a real number; a finite value is retained and an infinite value is assigned \(0\).
A real function \(X\) on a measure space \((\Omega ,\mu )\) is symmetric when \(X\) and \(-X\) are almost everywhere measurable and have the same distribution under \(\mu \).
A Rademacher variable on a measure space \((\Omega ,\mu )\) is a measurable \(r:\Omega \to \mathbb {R}\) satisfying \(r\in \{ -1,1\} \) almost everywhere and \(\mu (r=1)=\mu (r=-1)=1/2\).
Let \(F\) assign a real coefficient \(F(S)\) to every \(S\subseteq [n]\), and let \(X_i:\Omega \to \mathbb {R}\). Define \(F(X)(\omega )=\sum _{S\subseteq [n]}F(S)\prod _{i\in S}X_i(\omega )\).
A real function \(X\) on \((\Omega ,\mu )\) has atom lower bound \(\lambda \in \mathbb {R}\) when there exists a finite \(s\subseteq \mathbb {R}\) supporting \(X\) almost everywhere and every \(x\in s\) satisfies \(\mu (X=x)\ge \max (\lambda ,0)\).
Let \(q\gt 0\) and \(X\in L^q(\mu )\) on an arbitrary measure space. Its finite real norm satisfies \(\| X\| _q^q=\int |X|^q\, d\mu \).
Let \((\Omega ,\mu )\) be a probability space, let \(X\in L^2(\mu )\) satisfy \(\mathbb {E}X=0\), and let \(a,b\in \mathbb {R}\). Then \(\| a+bX\| _2^2=a^2+b^2\| X\| _2^2\).
Let \(r\) be a measurable random variable on a probability space, with \(r\in \{ -1,1\} \) almost surely and \(\Pr [r=1]=\Pr [r=-1]=1/2\). Its law is \(\tfrac 12\delta _1+\tfrac 12\delta _{-1}\).
Let \(X\) be centered on a probability space, let \(q\gt 2\), and suppose \(H_{2,q,\rho }(X)\). Then \(H_{q/(q-1),2,\rho }(X)\). Write \(H_{p,q,\rho }(X)\) for \(1\le p\le q\le \infty \), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for all \(a,b\in \mathbb {R}\).
Let \(X\) be a real function on an arbitrary measure space, suppose \(H_{p,q,\rho }(X)\), and let \(c\in \mathbb {R}\). Then \(H_{p,q,\rho }(cX)\). Write \(H_{p,q,\rho }(X)\) for \(1\le p\le q\le \infty \), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for all \(a,b\in \mathbb {R}\).
Let \(X\) be centered on a probability space, suppose \(H_{p,q,\rho }(X)\), and let \(0\le \sigma \le \rho \). Then \(H_{p,q,\sigma }(X)\), including extended exponents equal to infinity. Write \(H_{p,q,\rho }(X)\) for \(1\le p\le q\le \infty \), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for all \(a,b\in \mathbb {R}\).
Let \(\mu \) be a finite measure and \(X:\Omega \to \mathbb {R}\) measurable. If a finite set supports \(X\) almost everywhere and each listed value has mass at least \(\max (\lambda ,0)\) for some real \(\lambda \), then \(X\in L^p(\mu )\) for every \(p\in [0,\infty ]\), including essential boundedness.
Let \(X_1,\ldots ,X_n\) be mutually independent real random variables on a probability space, each belonging to \(L^4\). For arbitrary real coefficients \(F(S)\), the multilinear evaluation \(\sum _{S\subseteq [n]}F(S)\prod _{i\in S}X_i\) belongs to \(L^4\).
Let \(R,a,b,A,D,C\in \mathbb {R}\) satisfy \(R\ge 9\), \(a,b,C,D\ge 0\), \(A\le Ra^2\), \(D\le b^2\), and \(C^2\le AD\). Then \(A+6C+RD\le R(a+b)^2\).
Let \(s\) be a finite subset of an index set \(I\), let \(i\notin s\), let \(F\) assign real coefficients to finite subsets of \(I\), and let \(X_j:\Omega \to \mathbb {R}\). For every \(\omega \in \Omega \), \(\sum _{T\subseteq s\cup \{ i\} }F(T)\prod _{j\in T}X_j(\omega )=\sum _{T\subseteq s}F(T)\prod _{j\in T}X_j(\omega )+X_i(\omega )\sum _{T\subseteq s}F(T\cup \{ i\} )\prod _{j\in T}X_j(\omega )\).
Let \(\mu \) be a finite measure, let \(p,a,b\in \mathbb N\) satisfy \(a+b\le p\), and let real functions \(f,g\in L^p(\mu )\). Then \(f^ag^b\) is integrable.