Polynomial Bohnenblust--Hille Bounds and Hardy--Littlewood Inequalities for Finite Local Systems
Abstract
The complex polynomial Bohnenblust--Hille inequality states that, for a d-homogeneous polynomial P:ℂN→ℂ, P(z)=∑|α|=dcαzα, the coefficient norm (∑|α|=d|cα|pd)1/pd is bounded by Bdpol|P|∞, where pd=2d/(d+1) and Bdpol is independent of N. Here |P|∞ is the supremum norm on the unit polydisc. This inequality has a Boolean analogue for the Fourier--Walsh coefficients of functions of degree at most d, with the supremum taken over {−1,1}N. In this setting, Defant, Mastyło and Pérez obtained subexponential bounds, and Ivanisvili established polynomial growth. Further polynomial estimates were obtained by Slote, Tseng and Volberg in the Boolean setting and by Slote and Volberg for cyclic groups. We study these coefficient inequalities in a broader framework of finite systems of functions that encompasses and extends the Boolean and cyclic settings. This framework also admits a natural ℓp formulation, leading to Hardy--Littlewood-type inequalities. For orthonormal systems, we obtain polynomial bounds for the Bohnenblust--Hille constants and, for sufficiently large p, for the constants in these Hardy--Littlewood-type inequalities. In the Boolean case, the polynomial power is 5/2, with a cubic logarithmic factor.
Related
- Polynomial Bohnenblust--Hille bounds for product of cyclic groups
- Intrinsic Bohnenblust--Hille Inequalities for Local Qudit Systems
- Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems
- Support-Sensitive Bohnenblust-Hille Inequalities and Local Invariants on Hamming Schemes
- Polynomial growth of Bohnenblust--Hille constants on the Hamming cube
BibTeX
@misc{barbosa2026polynomial,
title = {{Polynomial Bohnenblust--Hille Bounds and Hardy--Littlewood Inequalities for Finite Local Systems}},
author = {Anderson Barbosa and Daniel Núñez-Alarcón and Daniel Pellegrino and Anselmo Raposo Jr},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15363},
eprint = {2610.15363},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15363},
}
RIS
TY - UNPB
TI - Polynomial Bohnenblust--Hille Bounds and Hardy--Littlewood Inequalities for Finite Local Systems
AU - Anderson Barbosa
AU - Daniel Núñez-Alarcón
AU - Daniel Pellegrino
AU - Anselmo Raposo Jr
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15363
UR - https://arxiv.org/abs/2610.15363
AB - The complex polynomial Bohnenblust--Hille inequality states that, for a d-homogeneous polynomial P:ℂ^N→ℂ, P(z)=∑_|α|=dc_αz^α, the coefficient norm (∑_|α|=d|c_α|^p_d)^1/p_d is bounded by B_d^pol|P|_∞, where p_d=2d/(d+1) and B_d^pol is independent of N. Here |P|_∞ is the supremum norm on the unit polydisc. This inequality has a Boolean analogue for the Fourier--Walsh coefficients of functions of degree at most d, with the supremum taken over {−1,1}^N. In this setting, Defant, Mastyło and Pérez obtained subexponential bounds, and Ivanisvili established polynomial growth. Further polynomial estimates were obtained by Slote, Tseng and Volberg in the Boolean setting and by Slote and Volberg for cyclic groups. We study these coefficient inequalities in a broader framework of finite systems of functions that encompasses and extends the Boolean and cyclic settings. This framework also admits a natural ℓ_p formulation, leading to Hardy--Littlewood-type inequalities. For orthonormal systems, we obtain polynomial bounds for the Bohnenblust--Hille constants and, for sufficiently large p, for the constants in these Hardy--Littlewood-type inequalities. In the Boolean case, the polynomial power is 5/2, with a cubic logarithmic factor.
ER -