# Square functions and the Hamming cube: Duality

## TL;DR

This paper establishes a new inequality relating the gradient and function moments on the Hamming cube for 1<p≤2, using duality between Euclidean square functions and Hamming cube gradient estimates.

## Contribution

It introduces a novel inequality connecting gradient norms and function moments on the Hamming cube, with a precise constant derived from hypergeometric functions.

## Key findings

- The inequality holds for all functions on the Hamming cube with 1<p≤2.
- The constant C(p) is characterized as the smallest positive zero of a confluent hypergeometric function.
- The approach reveals a duality between Euclidean square functions and Hamming cube gradient estimates.

## Abstract

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.