$f\in\mathcal{S}'^n$, if $D^\alpha f\in L^2$, prove $\xi^\alpha\hat f(\xi)\in L^2$

Question

Suppose $f\in\mathcal{S}'((\mathbb{R}^n))$, if $D^\alpha f\in L^2(\mathbb{R}^n)$, prove $\xi^\alpha\hat f(\xi)\in L^2(\mathbb{R}^n)$.

I know $F[D^\alpha f]=\xi^\alpha\hat f(\xi)$, so I want to prove $$ \int_{\mathbb{R}^n_x}\left(\int_{\mathbb{R}^n_y} e^{ixy}D^\alpha f(y)dy\right)^2dx<+\infty. $$ How to prove it?