Convolution theorem with distributions Let $\varphi:\mathbb{R}^n \to [0,\infty)$ be compactly supported and $C^{\infty}$. Define $k_s(x) = |x|^{-\alpha}$ for $x \in \mathbb{R}^n$, where $0 < \alpha < n$. I know that, as tempered distributions,
$$
\mathcal{F}(k_\alpha) = k_{n-\alpha}
$$
and
$$
\mathcal{F}(\varphi \ast k_\alpha) = \mathcal{F}(\varphi)\mathcal{F}(k_\alpha).
$$
Here $\mathcal{F}$ denotes the Fourier transform. 
Question Is the following true? If yes, how can it be proved? 
\begin{align*}
\int_{\mathbb{R}^n} | \mathcal{F}(\varphi \ast k_\alpha) |^2  dx = \int_{\mathbb{R}^n} | \mathcal{F}(\varphi)\mathcal{F}(k_\alpha) |^2 dx
\end{align*}
In my application, we can assume $2(\alpha-n) > -n$ so that the integral on the right is is finite, in case that helps. 
 A: The real question here is to show that $F(\varphi*u)=F(\varphi)\cdot F(u)$, as $L^2$ functions, for Schwartz functions $\varphi$ and suitable tempered distributions $u$, where $F(\varphi)\cdot F(u)$ is pointwise multiplication. For the latter to make sense, we need to have a prior condition on $u$ so that (for example) $F(u)$ is a locally integrable function, so has pointwise values a.e., and is completely described by those values (as opposed to Dirac $\delta$, for example). It suffices to have $u$ be in some Sobolev space, for example (all of which are inside the space of tempered distributions, and have locally $L^2$ Fourier transforms).
Then there is the question of a good definition/characterization of $\varphi*u$ for Schwartz functions and tempered distributions. One characterization is that it is a tempered distribution such that, for every Schwartz function $\psi$, $(\varphi*u)(\psi)=u(\varphi*\psi)$. For this to make sense, we need to know that $\varphi*\psi$ is Schwartz, which is indeed the case, by a variety of arguments.
Then, ignoring some signs (which disappear in the end),
$$
F(\varphi*u)(\psi) \;=\; (\varphi*u)(F\psi) \;=\; u(\varphi*F\psi)
\;=\; u(F(F\varphi\cdot \psi))
\;=\; (Fu)(F\varphi\cdot \psi)
\;=\; (F\varphi\cdot Fu)(\psi)
$$
by the usual characterization of multiplication of tempered distributions by Schwartz functions. This is the desired identity.
