Year: 2026
Type: preprint
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 $(r_1, \ldots, r_m, s')$-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 $A_{\vec{p}, (\vec{r}, s)}$, 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.
Source: arXiv (Cornell University)
Authors: Cao, Mingming, Du, Binyu, Yabuta, Kôzô
Institutions: Harbin Institute of Technology, Kwansei Gakuin University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221771379