Fourier level inequalities on the discrete cube via complex hypercontractivity and Riesz transforms

Eldan's level-1:3 inequality is a central tool in his second-order correlation inequalities for increasing Boolean functions. His proof is going to Gaussian space and uses an entropy estimate for second Hermite tensors. We give a direct proof on the discrete hypercube. For increasing real-valued functions f,g:\0,1\ⁿ→ R, we prove W₃[f,g]≥ -2W₁[f,g]log\!(√Var(f)Var(g)W₁[f,g]). The proof combines complex hypercontractivity with the discrete Riesz transforms. We also give a short proof of a level-$1:d$ inequality obtained by Keller and Kindler. As an application, we recover Eldan's correlation inequality with a slight improvement. If f,g:\0,1\ⁿ→\0,1\ are increasing and g is antipodal, then Cov(f,g)≥ 124\,∑ᵢInfᵢ[f]Infᵢ[g]√log\!(4e√Var(f)Var(g)/(∑ᵢInfᵢ[f]Infᵢ[g])), where the square-root logarithm cannot be improved as shown in the OR--majority example.