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 $Ω_n = \{ -1, 1\}^n$. Let $Δ=\sum_{j=1}^nD_j$ 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 \mat