$L^p$ 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 C^{1,1,1}: (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2 < \xi_3} \Big[ \prod_{j=1}^3 \hat{f}_j (\xi_j) \: e^{2 \pi i x \xi_j } \Big] \,d \vec{\xi} 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 \begin{align*} C^{1,1,-2}:\ & (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2 < -{\xi_3}/{2}}\Big[ \prod_{j=1}^3 \hat{f}_j (\xi_j) e^{2 \pi i x \xi_j } \Big] \,d \overrightarrow{\xi} \\ C^{1,1,1,-2}:\ & (f_1, f_2, f_3, f_4) \mapsto \int_{\xi_1 < \xi_2 < \xi_3 < -{\xi_4}/{2}} \Big[\prod_{j=1}^4 \hat{f}_j (\xi_j) e^{2 \pi i x \xi_j} \Big] d \overrightarrow{\xi} \end{align*} 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.