Generic Multilinear Multipliers Associated to Degenerate Simplexes

TL;DR

This paper proves boundedness of a class of degenerate multilinear multipliers associated with simplexes, extending their action to certain Lebesgue and Wiener spaces under specific conditions.

Contribution

It establishes the boundedness of generic degenerate trilinear simplex multipliers on mixed Lebesgue and Wiener spaces, generalizing previous results to degenerate cases.

Findings

Boundedness of the multiplier on specified function spaces.

Extension to Wiener spaces $W_p(R)$ under Hörmander-Mikhlin conditions.

Applicable for a range of $p_1, p_2, p_3$ satisfying given inequalities.

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*}