I'm with problems to prove an equality that appears in some proofs on book that I'm reading (An Introduction to Pseudo-Differential Operators).
We define the fourier transform of a function $f$ in $L^1(\Bbb{R}^n)$ by $$\widehat{f}(\xi)=(2\pi)^{-\frac{n}{2}}\int_{\Bbb{R}^n}e^{-ix\cdot\xi}f(x)dx,\quad \xi\in\Bbb{R}^n$$
First of all, let us define, for each $k=0,1,2,...$, $$\sigma_k(x,\xi)=\sigma(x,\xi)\varphi_k(\xi)$$ and $$K_k(x,z)=(2\pi)^{-\frac{n}{2}}\int_{\Bbb{R}^n}e^{iz\cdot\xi}\sigma_k(x,\xi)d\xi,$$ where $x,\xi,z\in\Bbb{R}^n$, $\sigma$ is a symbol in $S^m$ and $\{\varphi_k\}$ is the partition of unity.
The equality I need to prove:
$$(2\pi)^{-\frac{n}{2}}\int_{\Bbb{R}^n}e^{i\eta\cdot z}\frac{z^\mu}{\mu !}\partial_x^\mu K_k(x,z)dz=\frac{(-i)^{|\mu|}}{\mu !}(\partial_x^\mu\partial_\eta^\mu\sigma_k)(x,\eta),$$
where $\mu$ is any multi-indice.
The author says that we use the following proposition to obtain the result:
Proposition 2.2. For any funtion $\varphi$ in Schwarz space $S$, it holds:
$(\widehat{D^\alpha\varphi})(\xi)=\xi^\alpha\widehat{\varphi}(\xi),\quad\xi\in\Bbb{R}^n,\ \alpha\ \text{multi-indice};$
$(D^\beta\widehat{\varphi})(\xi)=(\widehat{(-x)^\beta\varphi})(\xi),\quad\xi\in\Bbb{R}^n,\ \beta\ \text{multi-indice}.$