Square functions and the Hamming cube: Duality

Abstract

For $1<p\le 2$, any $n\ge 1$ and any $f:\{-1,1{\}}^n\to \mathbb{R}$, we obtain $(\mathbb{E}|\nabla f{|}^p{)}^{1/p}\ge C(p)(\mathbb{E}|f{|}^p-|\mathbb{E}f{|}^p{)}^{1/p}$ 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.

18 pages