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Sparse domination and an extended Lp range for the trilinear Hilbert transform
Abstract
We establish sparse bounds for the trilinear Hilbert transform T(f1,f2,f3)(x):=p.v.∫ℝ f1(x−t)f2(x−2t)f3(x−3t) dt/t, with averaging exponents arbitrarily close to $2$ from above. As a consequence, we extend OpenAI's L3× L3× L3→ L1 bound (entry 086 in the OpenAI Research Catalog) to Lp1× Lp2× Lp3→ Lr throughout the range 2<p1,p2,p3<∞, 1/r=1/p1+1/p2+1/p3, 2/3<r<2, and obtain corresponding weighted estimates for weights in multilinear Muckenhoupt classes. Our proof relies on the normalized local estimate and stopping time argument developed in OpenAI's work.
Related
BibTeX
@misc{hu2026sparse,
title = {{Sparse domination and an extended $L^p$ range for the trilinear Hilbert transform}},
author = {Bingyang Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39096},
eprint = {2610.39096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39096},
}
RIS
TY - UNPB
TI - Sparse domination and an extended L^p range for the trilinear Hilbert transform
AU - Bingyang Hu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39096
UR - https://arxiv.org/abs/2610.39096
AB - We establish sparse bounds for the trilinear Hilbert transform T(f_1,f_2,f_3)(x):=p.v.∫_ℝ f_1(x−t)f_2(x−2t)f_3(x−3t) dt/t, with averaging exponents arbitrarily close to $2$ from above. As a consequence, we extend OpenAI's L^3× L^3× L^3→ L^1 bound (entry 086 in the OpenAI Research Catalog) to L^p_1× L^p_2× L^p_3→ L^r throughout the range 2<p_1,p_2,p_3<∞, 1/r=1/p_1+1/p_2+1/p_3, 2/3<r<2, and obtain corresponding weighted estimates for weights in multilinear Muckenhoupt classes. Our proof relies on the normalized local estimate and stopping time argument developed in OpenAI's work.
ER -