Generic Multilinear Multipliers Associated to Degenerate Simplexes

Abstract

For each $1\le p\le \infty$, 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 $a_1,a_2:{\mathbb{R}}^2\to \mathbb{C}$ satisfy the Hörmander-Mikhlin condition \begin{eqnarray*} \left| \partial^{\vecα} a_j \left(\vecξ\right) \right| \lesssim_{\vecα} \frac{1}{dist(\vecξ, Γ)^{|\vecα|}}~~~\forall \vecξ \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(ξ_1, ξ_2) a_2(ξ_2, ξ_3) \left[ \prod_{j=1}^3 \hat{f_j} (ξ_j) e^{2 πix ξ_j} \right] dξ_1 dξ_2 dξ_3 \end{eqnarray*} extends to a map $L^{p_1}(\mathbb{R})\times W_{p_2}(\mathbb{R})\times L^{p_3}(\mathbb{R})\to L^{\frac{1}{\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}}}(\mathbb{R})$ 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*}

References in corpus (1)

• Mixed Estimates for Degenerate Multilinear Operators Associated to Simplexes