The Endpoint Fractional Riesz Estimate on the Hamming Cube

Abstract

Let $1<p<2$ and $D_j$ be the discrete partial derivative on the Hamming cube ${\Omega }_n=\{-1,1{\}}^n$. Let $\Delta ={\sum }_{j=1}^n{D}_j$ be the discrete Laplacian.We prove the endpoint inequality \[ \left\|\left(\sum_{j=1}^n|D_jf|^2\right)^{1/2}\right\|_{p} \lesssim_p\|Δ^{1/p}f\|_{p}, \quad \forall f:Ω_n \to \mathbb C. \] This result answers the conjecture proposed by Naor, Eskenazis and Ivanisvili (see [BenEfraimLustPiquard, IvanisviliVolberg] or [Remark 45, EskenazisIvanisvili]). The proof relies heavily on the noncommutative semigroup BMO theory [JungeMei].