A spectral correlation inequality for increasing Boolean functions

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 $$\mathrm{C}\mathrm{o}\mathrm{v}(f,g)\ge 2\sum _{S\mathrm{\ne }\mathrm{\varnothing }}\mathrm{\mid }S\mathrm{\mid }\widehat{f}(S{)}^2\widehat{g}(S{)}^2$$ 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 $0\le \rho \le 1$, and every $f,g:\{0,1{\}}^n\to [0,1]$, the optimal constant $c_{\rho ,n}$ for which $$⟨f,T_{\rho }g⟩\ge c_{\rho ,n}\mathrm{\parallel }f\ast g{\mathrm{\parallel }}_2^2$$ holds for all such $f,g$ is $1$ for $0\le \rho \le 1\mathrm{/}2$, $(2(1-\rho ){)}^n$ for $1\mathrm{/}2<\rho <1$, and $0$ for $\rho =1$. Integrating this pointwise inequality yields, for $n\ge 1$, the slightly improved bound $$\mathrm{C}\mathrm{o}\mathrm{v}(f,g)\ge 4\cdot \frac{n+1}{2n}\sum _{S\mathrm{\ne }\mathrm{\varnothing }}\mathrm{\mid }S\mathrm{\mid }\widehat{f}(S{)}^2\widehat{g}(S{)}^2\mathrm{.}$$