I am working on a problem, and got stuck at this point.
Suppose that $u \in \mathcal{D}'(\mathbb{R}^n)$ is a distribution, $\phi_y \in \mathcal{D}(\mathbb{R}^n)$ for every $y \in \mathbb{R}^m$, and the map $\psi: y \mapsto u[\phi_y]$ is smooth, further $(D^{\alpha} \psi)(y) = (D^\alpha u)[ \phi_y]$ Can we say that $$\int_{\mathbb{R}^m} \psi(y) \ dy = u \left[ \int_{\mathbb{R}^m} \phi_y(x) \ dx \right]$$ I suspect that the answer is yes, but I am not sure how to proceed.