Generic Multilinear Multipliers Associated to Degenerate Simplexes

Abstract

For each $1 \leq p \leq \infty$, let $W_{p}(\mathbb{R}) = \left\{ f \in L^p(\mathbb{R}): \hat{f} \in L^{p^\prime}(\mathbb{R}) \right\}$ with norm $||f||_{W_{p}(\mathbb{R})} = ||\hat{f}||_{L^{p^\prime}(\mathbb{R})}$. Moreover, let $ \Gamma = \left\{ \xi_1 + \xi_2 =0\right\} \subset \mathbb{R}^2$ and $a_1,a_2 : \mathbb{R}^2 \rightarrow \mathbb{C}$ satisfy the H\"{o}rmander-Mikhlin condition \begin{eqnarray*} \left| \partial^{\vec{\alpha}} a_j \left(\vec{\xi}\right) \right| \lesssim_{\vec{\alpha}} \frac{1}{dist(\vec{\xi}, \Gamma)^{|\vec{\alpha}|}}~~~\forall \vec{\xi} \in \mathbb{R}^2, j \in \{1, 2\} \end{eqnarray*} for sufficiently many multi-indices $\vec{\alpha} \in (\mathbb{N} \bigcup \{0\})^2$. Our main result is that the generic degenerate trilinear simplex multiplier defined on $ \mathcal{S}^3(\mathbb{R})$ by \begin{eqnarray*} B[a_1, a_2] : (f_1, f_2, f_3) \rightarrow \int_{\mathbb{R}^3} a_1(\xi_1, \xi_2) a_2(\xi_2, \xi_3) \left[ \prod_{j=1}^3 \hat{f_j} (\xi_j) e^{2 \pi ix \xi_j} \right] d\xi_1 d\xi_2 d\xi_3 \end{eqnarray*} extends to a map $L^{p_1}(\mathbb{R}) \times W_{p_2}(\mathbb{R}) \times L^{p_3}(\mathbb{R}) \rightarrow L^{\frac{1}{\frac{1}{p_1} + \frac{1}{p _2} +\frac{1}{p_3}}}(\mathbb{R})$ provided \begin{eqnarray*} 1 < p_1, p_3 \leq \infty, \frac{1}{p_1} + \frac{1}{p_2} <1, \frac{1}{p_2} + \frac{1}{p_3} <1, 2 < p_2 <\infty. \end{eqnarray*}