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The Endpoint Fractional Riesz Estimate on the Hamming Cube

  • Zhendong Xu
  • Hao Zhang
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Abstract

Let 1<p<2 and Dj be the discrete partial derivative on the Hamming cube Ωn = { −1, 1}n. Let Δ=∑j=1nDj be the discrete Laplacian. We prove the endpoint inequality |(∑j=1n|Djf|2)1/2|p ≲p|Δ1/pf|p, ∀ f:Ωn → 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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Cite

BibTeXRIS
BibTeX
@misc{xu2026endpoint,
  title = {{The Endpoint Fractional Riesz Estimate on the Hamming Cube}},
  author = {Zhendong Xu and Hao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.03993},
  eprint = {2609.03993},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03993},
}
RIS
TY  - UNPB
TI  - The Endpoint Fractional Riesz Estimate on the Hamming Cube
AU  - Zhendong Xu
AU  - Hao Zhang
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.03993
UR  - https://arxiv.org/abs/2609.03993
AB  - Let 1<p<2 and D_j be the discrete partial derivative on the Hamming cube Ω_n = { −1, 1}^n. Let Δ=∑_j=1^nD_j be the discrete Laplacian. We prove the endpoint inequality |(∑_j=1^n|D_jf|^2)^1/2|_p ≲_p|Δ^1/pf|_p, ∀ f:Ω_n → 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].
ER  -