I have a doubt concearning the convergence of an integral.
Let $\mathscr{S}(\mathbb R^n)$ be the Schwartz space on $\mathbb R^n$. Given $u\in \mathscr{S}(\mathbb R^n)$ we have an well defined function:
$$(T_af)(x):=\int_{\mathbb R^n} e^{2\pi ix\cdot \xi} a(x, \xi)\widehat{u}(\xi)\ d\xi. $$
Indeed, this function belogs to $\mathscr{S}(\mathbb R^n)$.
Above $a\in C^\infty(\mathbb R^n\times \mathbb R^n)$ satisfies: $$|\partial^\alpha_x \partial^\beta_\xi a(x, \xi)|\leq C_{\alpha, \beta} (1+|\xi|^2)^{(m-|\beta|)/2},$$ for some constants $C_{\alpha, \beta}>0$ and some $m\in\mathbb R$ and $$\widehat{u}(\xi)=\int_{\mathbb R^n} e^{-2\pi iy\cdot \xi} u(y)\ dy,$$ is the Fourier transform of $u$.
What is the problem in rewriting $T_a$ as:
$$(T_af)(x)=\int_{\mathbb R^n} \int_{\mathbb R^n} e^{2\pi i(x-y)\cdot \xi} a(x, \xi)u(y)\ dy\ d\xi?$$
All I did was using the definition of $\widehat{u}$ in the first $T_af$ and added the two exponentials. It wasn't even needed changing integration order.
Thanks