题号:14616    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
Consider the Fourier transform. Let
$$
Q(g, f)(x):=\int_{\mathbb{R}^N} \int_{S^{N-1}} B\left(|x-y|, \frac{x-y}{|x-y|} \cdot \sigma\right) g\left(y^{\prime}\right) f\left(x^{\prime}\right) \mathrm{d} \sigma \mathrm{d} y
$$
where $B$ is a given two variable function, $S^{N-1}$ stands for the unit sphere in $\mathbb{R}^N$ and
$$
x^{\prime}:=\frac{x+y}{2}+\frac{|x-y| \sigma}{2} ; \quad y^{\prime}:=\frac{x+y}{2}-\frac{|x-y| \sigma}{2} .
$$

Then
$$
\widehat{Q(g, f)}(\xi)=(2 \pi)^{-N / 2} \int_{\mathbb{R}^N \times S^{N-1}} \hat{B}\left(|\eta|, \frac{\xi}{|\xi|} \cdot \sigma\right) \hat{g}\left(\xi^{-}+\eta\right) \hat{f}\left(\xi^{+}-\eta\right) \mathrm{d} \sigma \mathrm{d} \eta,
$$
where $\hat{B}(|\eta|, t):=\int_{\mathbb{R}^N} B(|q|, t) e^{-i q \cdot \eta} \mathrm{d} q, \xi^{ \pm}:=\frac{\xi \pm|\xi| \sigma}{2}$.
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