On a hybrid trilinear Hilbert transform
Abstract
In this paper we study the boundedness properties of a \emph{hybrid} trilinear Hilbert transform characterized by a \emph{dual regime} which simultaneously exhibits 1) \emph{resonant} (zero curvature) features, \emph{i.e.}, modulation invariance properties, and 2) \emph{non-resonant} (non-zero curvature) features. Specifically, given a∈(0,∞)∖ {1,2} and λ⃗:=(λ1,λ2,λ3,λ4)∈ (ℝ∖{0})4 with λ1≠ λ2 and λ3≠ λ4, we provide wide-range Lp bounds for the trilinear operator HTHTaλ⃗ (f1,f2,f3)(x):= p.v. ∫ℝ f1(x−λ1 t) f2(x−λ2 t−λ3 ta) f3(x−λ4 ta) dt/t, x ∈ ℝ . By combining variable multiresolution, multiscale wave-packet analysis, and Rank II LGC method, our present work establishes a common framework where the diverse methodologies of all of the following naturally converge: i) the maximal truncated bilinear Hilbert transform, ii) the Biest operator, iii) the hybrid bilinear Hilbert-Carleson operator, and iv) the non-resonant (curved) trilinear Hilbert transform. Ultimately, this work reveals that the hybrid trilinear transform sits on a critical intermediate scale between the purely non-resonant case and the purely resonant regime -- the latter remaining a challenging, major (open) problem in harmonic analysis.
Related
- Sparse domination and an extended Lp range for the trilinear Hilbert transform
- Sparse domination and the bi-parameter Hilbert transform
- A curved three-dimensional simplex Hilbert transform
- Optimal distributional estimates of the multiple Hilbert transform
- A Riemann-Hilbert approach to the computation of transform pairs
BibTeX
@misc{benea2026hybrid,
title = {{On a hybrid trilinear Hilbert transform}},
author = {Cristina Benea and Frédéric Bernicot and Victor Lie},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02167},
eprint = {2611.02167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02167},
}
RIS
TY - UNPB
TI - On a hybrid trilinear Hilbert transform
AU - Cristina Benea
AU - Frédéric Bernicot
AU - Victor Lie
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02167
UR - https://arxiv.org/abs/2611.02167
AB - In this paper we study the boundedness properties of a \emph{hybrid} trilinear Hilbert transform characterized by a \emph{dual regime} which simultaneously exhibits 1) \emph{resonant} (zero curvature) features, \emph{i.e.}, modulation invariance properties, and 2) \emph{non-resonant} (non-zero curvature) features. Specifically, given a∈(0,∞)∖ {1,2} and λ⃗:=(λ_1,λ_2,λ_3,λ_4)∈ (ℝ∖{0})^4 with λ_1≠ λ_2 and λ_3≠ λ_4, we provide wide-range L^p bounds for the trilinear operator HTHT^a_λ⃗ (f_1,f_2,f_3)(x):= p.v. ∫_ℝ f_1(x−λ_1 t) f_2(x−λ_2 t−λ_3 t^a) f_3(x−λ_4 t^a) dt/t, x ∈ ℝ . By combining variable multiresolution, multiscale wave-packet analysis, and Rank II LGC method, our present work establishes a common framework where the diverse methodologies of all of the following naturally converge: i) the maximal truncated bilinear Hilbert transform, ii) the Biest operator, iii) the hybrid bilinear Hilbert-Carleson operator, and iv) the non-resonant (curved) trilinear Hilbert transform. Ultimately, this work reveals that the hybrid trilinear transform sits on a critical intermediate scale between the purely non-resonant case and the purely resonant regime -- the latter remaining a challenging, major (open) problem in harmonic analysis.
ER -