The following argument is quoted from a book about Pseudodifferential operator. I am confused about the estimate of Fourier transform of a compactly supported function.
For a smooth function $p(x,\xi)$ defined on $\mathbb{R}^n \times \mathbb{R}^n,$ satisfying the estimate \begin{equation*} |\partial_x^\beta \partial_\xi^\alpha p(x,\xi) | \leq C_{\alpha,\beta} (1+|\xi|)^{m-|\alpha|}, \ \forall (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n, \end{equation*} and $C_{\alpha,\beta}$ is a constant number only depending on $\alpha,\beta.$ We assume further $p(x,\xi)$ has compact support in $x,$ that is, there exists $K$ independent on $\xi,$ such that $p(x,\xi)=0$ when $|x|\geq K.$
Now consider Fourier transform of $p(x,\xi)$ with respect to $\xi,$ the Fourier dual variable is $\eta \in \mathbb{R}^n,$ the author said that "concerning the support of $p(x,\xi)$, we have the following estimate" \begin{equation*} \mathcal{F}_{x \rightarrow \eta}(p)(\eta-\xi,\xi) \leq C (1+|\xi|)^m(1+|\eta-\xi|)^{-h}, \end{equation*} for all $h \geq 0,$ and $C$ is a positive number independent on $h.$
I have no idea how to deduce the estimate. Can anyone help me?