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 \begin{align*} \hspace{5mm} C^{1,1,1}:& (f_1, f_2, f_3) \mapsto \int_{ξ_1 < ξ_2< ξ_3} \left[ \prod_{j=1}^3 \hat{f}_j (ξ_j) e^{2 πi x ξ_j } \right] d \vecξ \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_{ξ_1 <ξ_2 < -\frac{ξ_3}{2}}\left[ \prod_{j=1}^3 \hat{f}_j (ξ_j) e^{2 πi x ξ_j } \right] d \vecξ \end{align*} \begin{align*} C^{1,1,1,-2}:& (f_1, f_2, f_3, f_4) \mapsto \int_{ξ_1 <ξ_2 < ξ_3< -\frac{ξ_4}{2}} \left[\prod_{j=1}^4 \hat{f}_j (ξ_j) e^{2 πi x ξ_j} \right] d \vecξ. \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.