Multi-scale analysis for multilinear operators

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 $(r_1,\dots ,r_m,s^{\mathrm{\prime }})$-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.