题号:14614    题型:解答题    来源:2024年丘成桐大学生数学竞赛(数学分析与微分方程类)-无答案
Let $\psi(\xi) \in C_c^{\infty}(\mathbb{R})$ be smooth and has compact support. Let $\psi(\xi)=0, \forall|\xi| \geq 1$. Let $f_1(\xi), f_2(\xi) \in C_c^{\infty}(\mathbb{R})$, i.e. $f_1, f_2$ ar e smooth and have compact support. Let $u_i: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{C}$, $i=1,2$, be defined as
$$
\begin{gathered}
u_1\left(x_1, x_2\right):=\int_{\mathbb{R}} \psi(\xi) f_1(\xi) e^{i \xi x_1} e^{i \xi^2 x_2} \mathrm{~d} \xi, \\
u_2\left(x_1, x_2\right):=\int_{\mathbb{R}} \psi(\eta-10) f_2(\eta) e^{i \eta x_1} e^{i \eta^2 x_2} \mathrm{~d} \eta .
\end{gathered}
$$

Prove there exists a constant $C$, which may depend on $\psi$, but does not depend on $f_1, f_2$, so that
$$
\left\|u_1 u_2\right\|_{L^2\left(\mathbb{R}^2\right)} \leq C\left\|f_1\right\|_{L^2(\mathbb{R})}\left\|f_2\right\|_{L^2(\mathbb{R})} .
$$
(Hint: One may try to use Plancherel Theorem. It may be usef ul to observe that if one let $H(\xi, \eta)=f_1(\xi) f_2(\eta)$, then $\|H\|_{L^2\left(\mathbb{R}^2\right)}$ are also bounded by $\left\|f_1\right\|_{L^2(\mathbb{R})}\left\|f_2\right\|_{L^2(\mathbb{R})}$
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