Multi-linear multipliers associated to simplexes of arbitrary length

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 $n=2$).

52 pages, 6 figures