Multi-scale analysis for multilinear operators
Abstract
We develop a unified multi-scale framework for sparse domination of multilinear operators beyond the multilinear Calderón--Zygmund theory. More precisely, under mild hypotheses on a support condition, weak type and restricted strong type uniform bounds for finite partial sums, and single-scale improving and regularity estimates, we establish (r1, …, rm, s')-sparse form bound for a family of localized m-linear operators uniformly over all finite partial sums. This sparse bound yields weighted norm inequalities for the corresponding multilinear Muckenhoupt classes Ap⃗, (r⃗, s), including the quasi-Banach range. As applications, we determine the optimal open sparse range for multilinear rough singular integrals, and characterize sparse domination for bilinear pseudo-differential operators. Further applications include sparse bounds for bilinear oscillatory Fourier multipliers, bilinear spherical maximal operators, and bilinear Bochner--Riesz means.
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BibTeX
@misc{cao2026multiscale,
title = {{Multi-scale analysis for multilinear operators}},
author = {Mingming Cao and Binyu Du and Kôzô Yabuta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16088},
eprint = {2610.16088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16088},
}
RIS
TY - UNPB
TI - Multi-scale analysis for multilinear operators
AU - Mingming Cao
AU - Binyu Du
AU - Kôzô Yabuta
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16088
UR - https://arxiv.org/abs/2610.16088
AB - We develop a unified multi-scale framework for sparse domination of multilinear operators beyond the multilinear Calderón--Zygmund theory. More precisely, under mild hypotheses on a support condition, weak type and restricted strong type uniform bounds for finite partial sums, and single-scale improving and regularity estimates, we establish (r_1, …, r_m, s')-sparse form bound for a family of localized m-linear operators uniformly over all finite partial sums. This sparse bound yields weighted norm inequalities for the corresponding multilinear Muckenhoupt classes A_p⃗, (r⃗, s), including the quasi-Banach range. As applications, we determine the optimal open sparse range for multilinear rough singular integrals, and characterize sparse domination for bilinear pseudo-differential operators. Further applications include sparse bounds for bilinear oscillatory Fourier multipliers, bilinear spherical maximal operators, and bilinear Bochner--Riesz means.
ER -