Non-Degenerate Multilinear Singular Multipliers with Fractional Rank
Jianghao Zhang
TL;DR
This work advances the theory of multilinear singular multipliers $T_{\\mathfrak{m}}$ acting on $n-1$ inputs in $\\mathbb{R}^d$ whose symbol decays away from an $m$-dimensional subspace $\\Gamma$, with rank $m/d$ governing complexity. The authors develop a time-frequency framework combining Whitney-type discretization, tensorization of the symbol, and a refined tree-based organization (including new Type I/II non-degeneracy conditions) to obtain $L^p$ bounds in three regimes, extending prior fractional-rank results and addressing large-rank scenarios. The approach relies on discrete models via vector rectangles and wave packets, plus Katz–Tao type counting lemmas for tree configurations, and culminates in interpolation to full ranges of exponents, including restricted-weak-type arguments with exceptional sets. These contributions broaden multilinear multiplier bounds in higher dimensions and offer a robust toolkit for handling fractional-rank degeneracies with potential PDE applications. Overall, the paper provides a comprehensive, generic strategy for establishing sharp $L^p$ bounds for a broad class of singular multilinear operators tied to fractional rank.
Abstract
We establish $L^p$ estimates for multilinear multipliers acting on $(n-1)$-tuples of functions on $\mathbb{R}^d$. We assume that the multiplier satisfies symbol estimates outside a linear subspace of dimension $m$. The difficulty of proving $L^p$ bounds increases with the rank $\frac{m}{d}$, and our focus is on the fractional rank case $\frac{m}{d}<\frac{n}{2}\leq \lceil \frac{m}{d}\rceil$.