\(L^p\) estimates for semi-degenerate simplex multipliers (Q2177513)

Summary: \textit{C. Muscalu} et al. [Math. Ann. 329, No. 3, 401--426 (2004; Zbl 1073.42009); ibid. 329, No. 3, 427--461 (2004; Zbl 1073.42010)] 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 \[ C^{1,1,-2}: (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2 < -{\xi_3}/{2}}\Big[ \prod_{j=1}^3 \hat{f}_j (\xi_j) \: e^{2 \pi i x \xi_j } \Big] \,d \vec{\xi} \] \[ C^{1,1,1,-2}:(f_1, f_2, f_3, f_4) \mapsto \int_{\xi_1 < \xi_2 < \xi_3 < -{\xi_4}/{2}} \Big[\prod_{j=1}^4 \hat{f}_j (\xi_j) \: e^{2 \pi i x \xi_j} \Big] \,d \vec{\xi} \] 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.