A spectral correlation inequality for increasing Boolean functions

Abstract

Talagrand&#39;s correlation inequality provides a quantitative strengthening of the Harris--Kleitman inequality for increasing Boolean functions. Motivated by a Fourier-analytic conjecture of Friedgut, Kahn, Kalai, and Keller, we prove that $$ \mathrm{Cov}(f,g)\ge 2\sum_{S\neq\emptyset}|S|\hat f(S)^2\hat g(S)^2 $$ holds for all increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$. The proof combines the reverse Bonami--Beckner inequality with Young&#39;s convolution inequality. We also establish a sharp pointwise inequality: for every $n\ge1$, every $0\leρ\le1$, and every $f,g:\{0,1\}^n\to[0,1]$, the optimal constant $c_{ρ,n}$ for which $$ \left\langle f,T_ρg \right\rangle\ge c_{ρ,n}\|f*g\|_2^2 $$ holds for all such $f,g$ is $1$ for $0\leρ\le1/2$, $(2(1-ρ))^n$ for $1/2<ρ<1$, and $0$ for $ρ=1$. Integrating this pointwise inequality yields, for $n\ge1$, the slightly improved bound $$ \mathrm{Cov}(f,g)\ge 4\cdot\frac{n+1}{2n}\sum_{S\neq\emptyset}|S|\hat f(S)^2\hat g(S)^2. $$