Sparse Kahane--Salem--Zygmund Forms and Weighted Hardy--Littlewood Inequalities Across the Critical Endpoint
Abstract
We study sparse Kahane--Salem--Zygmund constructions and weighted Hardy--Littlewood inequalities for homogeneous polynomials. For supports of cardinality nd+o(1), we determine the sharp power of n governing the smallest norm of a unimodular m-linear form on ℓp1n×⋯×ℓpmn; in the diagonal case, this yields the missing polynomial growth exponent in the coefficient-versus-supremum norm problem for 2≤ p≤ m and 2≤ r≤∞. We then introduce a diagonal weighted Hardy--Littlewood functional which, on s=p≥ m, agrees exactly with the classical Hardy--Littlewood coefficient norm with the same optimal constant. We determine the optimal diagonal weight exponent for 2≤ p≤ m and 1≤ q≤2, on the full critical line p=m, and on a sharp part of the region q>2; at q=∞ the optimal weight exponent is obtained for every 2≤ p≤ m. The sparse coefficient estimates provide the matching dimensional obstructions.
Related
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BibTeX
@misc{barbosa2026sparse,
title = {{Sparse Kahane--Salem--Zygmund Forms and Weighted Hardy--Littlewood Inequalities Across the Critical Endpoint}},
author = {Anderson Barbosa and Daniel Núñez-Alarcón and Anselmo Raposo and Joedson Santos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01500},
eprint = {2610.01500},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01500},
}
RIS
TY - UNPB
TI - Sparse Kahane--Salem--Zygmund Forms and Weighted Hardy--Littlewood Inequalities Across the Critical Endpoint
AU - Anderson Barbosa
AU - Daniel Núñez-Alarcón
AU - Anselmo Raposo
AU - Joedson Santos
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01500
UR - https://arxiv.org/abs/2610.01500
AB - We study sparse Kahane--Salem--Zygmund constructions and weighted Hardy--Littlewood inequalities for homogeneous polynomials. For supports of cardinality n^d+o(1), we determine the sharp power of n governing the smallest norm of a unimodular m-linear form on ℓ_p_1^n×⋯×ℓ_p_m^n; in the diagonal case, this yields the missing polynomial growth exponent in the coefficient-versus-supremum norm problem for 2≤ p≤ m and 2≤ r≤∞. We then introduce a diagonal weighted Hardy--Littlewood functional which, on s=p≥ m, agrees exactly with the classical Hardy--Littlewood coefficient norm with the same optimal constant. We determine the optimal diagonal weight exponent for 2≤ p≤ m and 1≤ q≤2, on the full critical line p=m, and on a sharp part of the region q>2; at q=∞ the optimal weight exponent is obtained for every 2≤ p≤ m. The sparse coefficient estimates provide the matching dimensional obstructions.
ER -