Poincar\'e inequality 3/2 on the Hamming cube

TL;DR

This paper establishes a new Poincaré inequality of order 3/2 on the Hamming cube, relating the expectation of a complex power of a function and its gradient, with implications for analysis on discrete structures.

Contribution

It introduces a novel inequality involving the 3/2 power and the gradient on the Hamming cube, expanding the understanding of functional inequalities in discrete settings.

Findings

Proves the inequality for all functions on the Hamming cube.

Connects the inequality to complex analysis via principal branch of power.

Provides a new tool for analysis on discrete hypercube structures.

Abstract

For any $n \geq 1$, and any $f :\{-1,1\}^{n} \to \mathbb{R}$ we have $ \Re\, \mathbb{E}\, (f + i\, |\nabla f|)^{3/2} \leq \Re\, (\mathbb{E}f)^{3/2}, $ where $z^{3/2}$ for $z=x+iy$ is taken with principal branch and $\Re$ denotes the real part.