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arXiv · 2609.03993

The Endpoint Fractional Riesz Estimate on the Hamming Cube

Zhendong Xu·Hao Zhang
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Abstract

Let $1<p<2$ and $D_j$ be the discrete partial derivative on the Hamming cube $\Omega_n = \{ -1, 1\}^n$. Let $\Delta=\sum_{j=1}^nD_j$ be the discrete Laplacian.We prove the endpoint inequality \[ \left\|\left(\sum_{j=1}^n|D_jf|^2\right)^{1/2}\right\|_{p} \lesssim_p\|\Delta^{1/p}f\|_{p}, \quad \forall f:\Omega_n \to \mathbb C. \] 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].

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Zhendong Xu, Hao Zhang. 2026-09-03. The Endpoint Fractional Riesz Estimate on the Hamming Cube. https://arxiv.org/abs/2609.03993

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