PreprintCC BY 4.0arXiv
Sharp Fractional Riesz Estimates on the Hypercube
Abstract
Let Ωn={−1,1}n be the n-dimensional hypercube equipped with the normalized uniform measure, let ∇ be the Walsh gradient and let Δ be the Walsh Laplacian. For every 1<p≤ 2 we prove the following estimate |∇ f|Lp(Ωn;ℓ2n) ≤ cabs(p−1)−2|Δ1/pf|Lp(Ωn). The exponent 1/p is optimal, thus this settles the open problem on the sharp fractional Riesz estimate by Efraim and Lust-Piquard \cite{E-LP2008} which was subsequently highlighted by Ivanisvili and Volberg \cite{I-V2022}. We also establish the higher-order counterpart. As applications of our results, we obtain simpler proofs of the optimal short-time estimate for ∇ e−tΔ, and the Bernstein-Markov type inequality for d-bounded degree functions.
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BibTeX
@misc{jiao2026sharp,
title = {{Sharp Fractional Riesz Estimates on the Hypercube}},
author = {Yong Jiao and Sijie Luo and Dmitriy Zanin and Dejian Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09040},
eprint = {2609.09040},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09040},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Fractional Riesz Estimates on the Hypercube
AU - Yong Jiao
AU - Sijie Luo
AU - Dmitriy Zanin
AU - Dejian Zhou
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09040
UR - https://arxiv.org/abs/2609.09040
AB - Let Ω_n={−1,1}^n be the n-dimensional hypercube equipped with the normalized uniform measure, let ∇ be the Walsh gradient and let Δ be the Walsh Laplacian. For every 1<p≤ 2 we prove the following estimate |∇ f|_L_p(Ω_n;ℓ_2^n) ≤ c_abs(p−1)^−2|Δ^1/pf|_L_p(Ω_n). The exponent 1/p is optimal, thus this settles the open problem on the sharp fractional Riesz estimate by Efraim and Lust-Piquard \cite{E-LP2008} which was subsequently highlighted by Ivanisvili and Volberg \cite{I-V2022}. We also establish the higher-order counterpart. As applications of our results, we obtain simpler proofs of the optimal short-time estimate for ∇ e^−tΔ, and the Bernstein-Markov type inequality for d-bounded degree functions.
ER -