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arXiv 1609.05946math.CA
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与退化单形相关的一般多线性乘数

Generic Multilinear Multipliers Associated to Degenerate Simplexes

Robert M. Kesler

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摘要：

对于每个$1 \leq p \leq \infty$，设$W_{p}(\mathbb{R}) = \left\{ f \in L^p(\mathbb{R}): \hat{f} \in L^{p^\prime}(\mathbb{R}) \right\}$具有范数$||f||_{W_{p}(\mathbb{R})} = ||\hat{f}||_{L^{p^\prime}(\mathbb{R})}$。此外，对于足够多的多指标$\vecα \in (\mathbb{N} \bigcup \{0\})^2$，令$ Γ= \left\{ ξ_1 + ξ_2 =0\right\} \subset \mathbb{R}^2$和$a_1,a_2 : \mathbb{R}^2 \rightarrow \mathbb{C}$满足Hörmander-Mikhlin条件\begin{eqnarray*} \left| \partial^{\vecα} a_j \left(\vecξ\right) \right| \lesssim_{\vecα} \frac{1}{dist(\vecξ, Γ)^{|\vecα|}}~~~\forall \vecξ \in \mathbb{R}^2, j \in \{1, 2\} \end{eqnarray*}。我们的主要结果是：在\begin{eqnarray*} 1 < p_1, p_3 \leq \infty, \frac{1}{p_1} + \frac{1}{p_2} <1, \frac{1}{p_2} + \frac{1}{p_3} <1, 2 < p_2 <\infty. \end{eqnarray*}的条件下，由\begin{eqnarray*} B[a_1, a_2] : (f_1, f_2, f_3) \rightarrow \int_{\mathbb{R}^3} a_1(ξ_1, ξ_2) a_2(ξ_2, ξ_3) \left[ \prod_{j=1}^3 \hat{f_j} (ξ_j) e^{2 πix ξ_j} \right] dξ_1 dξ_2 dξ_3 \end{eqnarray*}定义在$ \mathcal{S}^3(\mathbb{R})$上的一般退化三线性单纯形乘子扩张到一个映射$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})$

英文摘要：

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 $ Γ= \left\{ ξ_1 + ξ_2 =0\right\} \subset \mathbb{R}^2$ and $a_1,a_2 : \mathbb{R}^2 \rightarrow \mathbb{C}$ 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 $\vecα \in (\mathbb{N} \bigcup \{0\})^2$. 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 $L^{p_1}(\mathbb{R}) \times W_{p_2}(\mathbb{R}) \times L^{p_3}(\mathbb{R}) \rightarrow L^{\frac{1}{\frac{1}{p_1} + \frac{1}{p _2} +\frac{1}{p_3}}}(\mathbb{R})$ 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*}

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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