Square functions and the Hamming cube: Duality

For $1<p\leq 2$, any $n\geq 1$ and any $f:\{-1,1\}^{n} \to \mathbb{R}$, we obtain $(\mathbb{E} |\nabla f|^{p})^{1/p} \geq C(p)(\mathbb{E}|f|^{p} - |\mathbb{E}f|^{p})^{1/p}$ where $C(p)$ is the smallest positive zero of the confluent hypergeometric function ${}_{1}F_{1}(\frac{p}{2(1-p)}, \frac{1}{2}, \frac{x^{2}}{2})$. 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.