275 Boolean Analysis — Two Point

275.1 Overview

Symmetric real-power inequalities establish the one-bit hypercontractive bound for finite exponents between one and two. The statements include the normalized scalar comparison and its affine-function consequence.

275.2 Declarations

Lemma 275.1 Symmetric power average is at least one

✓

Let \(p \ge 1\) be a real number and let \(b \in [0,1]\). Then

\[ 1 \; \le \; \frac{(1+b)^p + (1-b)^p}{2}. \]

Lemma 275.2 Monotonicity of symmetric convex sums

✓

Let \(f\colon \mathbb {R}\to \mathbb {R}\) be convex on the half-line \([0,\infty )\), and let \(x,y\) be real numbers with \(0 \le x \le y \le 1\). Then

\[ f(1+x) + f(1-x) \; \le \; f(1+y) + f(1-y). \]

Lemma 275.3 Antitonicity in the exponent for nonpositive powers

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Let \(0 \lt x \lt 1\), and let \(p \le q \le 0\) be real exponents. Then

\[ (1+x)^p + (1-x)^p \; \ge \; (1+x)^q + (1-x)^q. \]

Lemma 275.4 A two-point convexity inequality

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Let \(r,s\in \mathbb {R}\) satisfy \(0\le r\le s\le 1\), let \(t\in [0,1]\), and set \(c=\sqrt{r/s}\) when \(s\gt 0\) and \(c=0\) when \(s=0\). With \(0^0=1\), one has

\[ (1+t)^r-(1-t)^r\ge c\bigl((1+ct)^s-(1-ct)^s\bigr). \]

Lemma 275.5 Integrated two-point hypercontractive inequality

✓

Let \(p,q,b\in \mathbb {R}\) satisfy \(1\le p\le q\le 2\) and \(0\le b\le 1\). Set \(\rho =\sqrt{(p-1)/(q-1)}\) when \(q\gt 1\) and \(\rho =0\) when \(q=1\). Then

\[ \frac{(1+\rho b)^q+(1-\rho b)^q-2}{q} \le \frac{(1+b)^p+(1-b)^p-2}{p}. \]

Lemma 275.6 Tangent line inequality for \(x^r\) at \(x=1\)

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For every real number \(x \ge 0\) and every exponent \(r \ge 1\), the power \(x^r\) lies above its tangent line at \(x = 1\); that is,

\[ x^r \ge 1 + r\, (x - 1). \]

Theorem 275.7 General two-point inequality, unit case

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Let \(p,q,b\in \mathbb {R}\) satisfy \(1\le p\le q\le 2\) and \(0\le b\le 1\). Set \(\rho =\sqrt{(p-1)/(q-1)}\) when \(q\gt 1\) and \(\rho =0\) when \(q=1\). Then

\[ \frac{(1+\rho b)^q+(1-\rho b)^q}{2} \le \left(\frac{(1+b)^p+(1-b)^p}{2}\right)^{q/p}. \]

Theorem 275.8 One-bit low-norms hypercontractivity

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Let \(1 \lt p \le q \le 2\), let \(f : \{ 0,1\} ^1 \to \mathbb {R}\) be a Boolean function of arity \(1\), and set \(\rho = \sqrt{(p-1)/(q-1)}\). Then, with expectations taken under the uniform measure and \(T_\rho \) the noise operator,

\[ \bigl(\mathbb {E}\bigl[\, \left\lvert T_\rho f\right\rvert ^q\, \bigr]\bigr)^{1/q} \; \le \; \bigl(\mathbb {E}\bigl[\, \left\lvert f\right\rvert ^p\, \bigr]\bigr)^{1/p}. \]

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.