A spectral correlation inequality for increasing Boolean functions

Abstract

Talagrand'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 $$\mathrm{Cov}(f,g)\ge 2\sum _{S\ne \varnothing }|S|\hat{f}(S{)}^2\hat{g}(S{)}^2$$ holds for all increasing Boolean functions $f,g:\{0,1{\}}^n\to \{0,1\}$. The proof combines the reverse Bonami--Beckner inequality with Young's convolution inequality. We also establish a sharp pointwise inequality: for every $n\ge 1$, every $0\le \ddot{I}\le 1$, and every $f,g:\{0,1{\}}^n\to [0,1]$, the optimal constant $c_{\ddot{I},n}$ for which $$⟨f,T_{\ddot{I}}g⟩\ge c_{\ddot{I},n}\Vert f\ast g{\Vert }_2^2$$ holds for all such $f,g$ is $1$ for $0\le \ddot{I}\le 1/2$, $(2(1-\ddot{I}){)}^n$ for $1/2<\ddot{I}<1$, and $0$ for $\ddot{I}=1$. Integrating this pointwise inequality yields, for $n\ge 1$, the slightly improved bound $$\mathrm{Cov}(f,g)\ge 4\cdot \frac{n+1}{2n}\sum _{S\ne \varnothing }|S|\hat{f}(S{)}^2\hat{g}(S{)}^2.$$

13 pages, comments are welcome!