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*}
Categories: math.CA