Estimate derivatives in terms of derivatives of the Fourier transform.

Question

Let us suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a smooth function. Furthermore, for every $\alpha$ multi-index, there exists $C_\alpha > 0$ such that $$ |D^\alpha f(\xi)| \leq \frac{C_\alpha}{(1+|\xi|)^{|\alpha|}}. $$ Does it follow that, for every $\alpha$, there exists $C'_\alpha > 0$ such that $$ |D^\alpha (\mathcal{F}^{-1}(f))(x)| \leq \frac{C'_\alpha}{|x|^{n+|\alpha|}} $$ where $\mathcal{F}^{-1}$ is the inverse Fourier transform (which exists since $f \in \mathcal{S}'$)? I tried to do it using the definition, but it is really messed up because $\mathcal{F}^{-1}$ is in general in $\mathcal{S}'$. For instance, if $f$ is a constant function, then its inverse transform is a dirac $\delta$, then I should give it a pointwise meaning, and I don't know when this is possible. Any help would be really appreciated.

Answer 1

This settles it. See Theorem 9, it also settles regularity issues.