Multi-linear multipliers associated to simplexes of arbitrary length

TL;DR

This paper proves boundedness of a class of multilinear operators associated with simplexes of arbitrary length, extending previous results on bi-linear and bi-est operators to higher dimensions.

Contribution

It generalizes known boundedness results for bi-linear operators to n-linear operators associated with simplexes of any length.

Findings

Boundedness of n-linear operators with simplex characteristic functions from L^2 to L^{2/n}

Extension of Lacey-Thiele theorem to higher dimensions

Unification of bi-linear and multi-linear operator bounds

Abstract

In this article we prove that the $n$-linear operator whose symbol is the characteristic function of the simplex $\Delta_n = \xi_1 < ... < \xi_n$ is bounded from $L^2 \times ... \times L^2$ into $L^{2/n}$, generalizing in this way our previous work on the "bi-est" operator (which corresponds to the case $n=3$) as well as Lacey-Thiele theorem on the bi-linear Hilbert transform (which corresponds to the case $n=2$).