For $0<\alpha<n$ and $n\geq 2$ we define the *Riesz potential* by
$$
(I_\alpha f)(x) = \frac{1}{\gamma(\alpha)}
\int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}}\, dy\, ,
\quad
\text{where}
\quad
\gamma(\alpha)=
\frac{\pi^{\frac{n}{2}}\, 2^\alpha\,\Gamma\left(\frac{\alpha}{2}\right)}
{\Gamma\left(\frac{n-\alpha}{2}\right)}\, .
$$
The following result is well known and regarded as nearly obvious:

Theorem.If $\alpha,\beta>0$, $\alpha+\beta<n$, then $I_\alpha(I_\beta\varphi)=I_{\alpha+\beta}\varphi$ for $\varphi\in\mathscr S_n$.

Formally \begin{eqnarray*} (I_\alpha I_\beta\varphi)^\wedge(\xi) & = & \left((-\Delta)^{\alpha/2}(-\Delta)^{\beta/2}\varphi)\right)^\wedge(\xi) = (4\pi^2|\xi|^2)^{\alpha/2}(4\pi^2|\xi|^2)^{\beta/2}\hat{\varphi}(\xi)\\ & = & (4\pi^2|\xi|^2)^{\frac{\alpha+\beta}{2}}\hat{\varphi}(\xi) = \Big((-\Delta)^{\frac{\alpha+\beta}{2}}\varphi\Big)^\wedge(\xi) =(I_{\alpha+\beta}\varphi)^\wedge(\xi) \end{eqnarray*} and the result follows by taking the inverse Fourier transform.

Having this argument in mind, Stein on p.118 in his *Singular integrals and differentiable properties of functions* writes `deduction of this formula offer no difficulties'. But is it really a rigorous proof? The problem is that $I_\alpha\varphi$ is defined as a convolution with the distribution, $I_\alpha\varphi=u_\alpha*\varphi$, where $u_\alpha=\gamma(\alpha)^{-1}|x|^{\alpha-n}$. We know that in that case we can use the formula $(u_\alpha*\varphi)^\wedge=\hat{\varphi}\hat{u}_\alpha$. Unfortunately, for most of the functions $\varphi\in\mathscr S_n$, $u_\alpha*\varphi=I_\alpha\varphi\not\in\mathscr S_n$ so the formula
$$
(I_\alpha(I_\beta\varphi))^\wedge=(u_\beta*(u_\alpha*\varphi))^\wedge=
(u_\alpha*\varphi)^\wedge \hat{u}_\beta
$$
is not properly justified.

My question is: **Do you know a reference to a rigorous and detailed proof of the above fact?**

I will provide an answer to my question by showing how I prove it rigorously, but, perhaps my argument is overly complicated and I am simply not able to see obvious things. Also, I could not find a rigorous and elementary proof anywhere.