L^p Estimates for Semi-Degenerate Simplex Multipliers

Research paper by Robert M. Kesler

Indexed on: 19 Sep '16Published on: 19 Sep '16Published in: arXiv - Mathematics - Classical Analysis and ODEs


C. Muscalu, T. Tao, and C. Thiele prove $L^p$ estimates for a non-degenerate trilinear simplex multiplier called the Biest, which is defined for $(f_1, f_2, f_3) \in \mathcal{S}^3(\mathbb{R})$ by the map C^{1,1,1} : (f_1, f_2, f_3) \mapsto \int_{\xi_1 < \xi_2 < \xi_3} \hat{f}_1(\xi_1) \hat{f}_2(\xi_2) \hat{f}_3(\xi_3) e^{2 \pi i x ( \xi_1 + \xi_2 + \xi_3)} d\xi_1 d \xi_2 d\xi_3. Their methods automatically produce bounds for the collection of all non-degenerate trilinear simplex symbols. Our aim in this article is to prove $L^p$ estimates for a pair of so-called semi-degenerate simplex multipliers given by &&C^{1,1,-2}: (f_1, f_2, f_3) \mapsto \int_{\xi_1 <\xi_2 < \xi_3} \hat{f}_1(\xi_1) \hat{f}_2(\xi_2) \hat{f}_3(\xi_3) e^{2 \pi i x (\xi_1 + \xi_2 - 2 \xi_3)} d\xi_1 d \xi_2 d \xi_3 && C^{1,1,1,-2}: (f_1, f_2, f_3, f_4) \mapsto \int_{\xi_1 <\xi_2 < \xi_3< \xi_4} \hat{f}_1(\xi_1) \hat{f}_2(\xi_2) \hat{f}_3(\xi_3) \hat{f}_4(\xi_4) e^{2 \pi i x (\xi_1 + \xi_2 + \xi_3-2 \xi_4)} d\xi_1 d \xi_2 d \xi_3 d \xi_4 for which the non-degeneracy condition fails. We obtain as corollaries that $C^{1,1,-2}$ maps into $L^p(\mathbb{R})$ for all $1/2< p < \infty$ and $C^{1,1,1,-2}$ maps into $L^p(\mathbb{R})$ for all $1/3 < p < \infty$. Both target $L^p$ ranges are shown to be sharp.