半退化单纯形乘子的L^p估计
L^p Estimates for Semi-Degenerate Simplex Multipliers
摘要：
Muscalu，陶，和Thiele证明$L^p$估计的“Biest”算子定义的施瓦茨函数的地图\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*}通过时间频率参数，产生的所有乘数与非退化三线性单纯形符号的界限。在这篇文章中，我们证明了$L^p$估计的一对单纯形乘数的非退化条件失败，并定义在施瓦茨函数的地图\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*}我们的论点结合了标准的$\ell^2$为基础的能源与$\ell^1$为基础的能源，以使求和各种大小参数。因此，我们得到，$C^{1,1,-2}$映射到$L^p$的所有$1/2< p < \infty$和$C^{1,1,1,-2}$映射到$L^p$的所有$1/3 < p < \infty$。两个目标$L^p$范围均显示为锐利。
英文摘要：
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.
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