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Generic Trilinear Multipliers Associated to Degenerate Simplexes

Research paper by Robert M. Kesler

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



Abstract

For each $2 < p\leq \infty$, define $W_{p}(\mathbb{R}) = \left\{ f \in L^p(\mathbb{R}): \hat{f} \in L^{p^\prime}(\mathbb{R}) \right\}$ with norm $||f_2||_{W_{p_2}(\mathbb{R})} = ||\hat{f}_2||_{L^{p_2^\prime}(\mathbb{R})}$. Furthermore, 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 \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\}for sufficiently many multi-indices $\vec{\alpha} \in (\mathbb{N} \bigcup \{0\})^2$. Then we prove that the generic degenerate trilinear simplex multiplier defined for any $(f_1, f_2, f_3) \in \mathcal{S}^3(\mathbb{R})$ by the formula 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 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 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.