The Endpoint Fractional Riesz Estimate on the Hamming Cube

Let $1<p<2$ and $D_j$ be the discrete partial derivative on the Hamming cube ${\mathrm{\Omega }}_n=\{-1,1{\}}^n$. Let $\mathrm{\Delta }={\sum }_{j=1}^n{D}_j$ be the discrete Laplacian.We prove the endpoint inequality $${\parallel {\left(\sum _{j=1}^n\mathrm{\mid }{D}_{j}f{\mathrm{\mid }}^2\right)}^{1\mathrm{/}2}\parallel }_p{\lesssim }_p\mathrm{\parallel }{\mathrm{\Delta }}^{1\mathrm{/}p}f{\mathrm{\parallel }}_p,\mathrm{\forall }f:{\mathrm{\Omega }}_n\to \mathbb{C}\mathrm{.}$$ 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].