Revista Matemática Iberoamericana


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Volume 36, Issue 1, 2020, pp. 99–158
DOI: 10.4171/rmi/1123

Published online: 2019-09-26

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

Robert Kesler[1]

(1) Santa Monica, USA

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 \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} \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.

Keywords: Semi-degenerate simplex multipliers

Kesler Robert: $L^p$ estimates for semi-degenerate simplex multipliers. Rev. Mat. Iberoam. 36 (2020), 99-158. doi: 10.4171/rmi/1123