$L^p$ estimates for semi-degenerate simplex multipliers

Abstract

Muscalu, Tao, and Thiele prove $L^p$ estimates for the "Biest" operator defined on Schwartz functions by the map

via a time-frequency argument that produces bounds for all multipliers with non-degenerate trilinear simplex symbols. In this article we prove $L^p$ estimates for a pair of simplex multipliers defined on Schwartz functions by the maps

for which the non-degeneracy condition fails. Our argument combines the standard ${\ell }^2$-based energy with an ${\ell }^1$-based energy in order to enable summability over various size parameters. As a consequence, we obtain that $C^{1,1,-2}$ maps into $L^p$ for all $1/2<p<\infty$ and $C^{1,1,1,-2}$ maps into $L^p$ for all $1/3<p<\infty$. Both target $L^p$ ranges are shown to be sharp.