L^p Estimates for Semi-Degenerate Simplex Multipliers

Muscalu, Tao, and Thiele prove $L^p$ estimates for the "Biest" operator defined on Schwartz functions by the map \begin{align*} \hspace{5mm} C^{1,1,1}:& (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2< \xi_3} \left[ \prod_{j=1}^3 \hat{f}_j (\xi_j) e^{2 \pi i x \xi_j } \right] d \vec{\xi} \end{align*} 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 for which the non-degeneracy condition fails and which are defined on Schwartz functions by the maps \begin{align*} C^{1,1,-2}:& (f_1, f_2, f_3) \mapsto \int_{\xi_1 <\xi_2 < -\frac{\xi_3}{2}}\left[ \prod_{j=1}^3 \hat{f}_j (\xi_j) e^{2 \pi i x \xi_j } \right] d \vec{\xi} \end{align*} \begin{align*} C^{1,1,1,-2}:& (f_1, f_2, f_3, f_4) \mapsto \int_{\xi_1 <\xi_2 < \xi_3< -\frac{\xi_4}{2}} \left[\prod_{j=1}^4 \hat{f}_j (\xi_j) e^{2 \pi i x \xi_j} \right] d \vec{\xi}. \end{align*} 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.