Square functions and the Hamming cube: duality

For 1 < p <= 2, any n >= 1 and any f : {-1,1}(n) -> R, we obtain (E vertical bar Vf vertical bar(P))1/P >= C(P)(E vertical bar f vertical bar(p) - vertical bar Ef vertical bar(p))(1/P) where c(p) is the smallest positive zero of the confluent hypergeometric function F-1(1) (p/2(1-p), 1/2, x(2)/2). Our approach is based on a certain duality between the classical square function estimates on the Euclidean space and the gradient estimates on the Hamming cube.