Suppose $u\in \mathcal{D}'(\mathbb{R}^n\times\mathbb{R}^m)$.
In general, it is well known that we cannot define a restriction map $\mathcal{R}:\mathcal{D}'(\mathbb{R}^n\times\mathbb{R}^m)\rightarrow \mathcal{D}'(\mathbb{R}^n)$ which generalises the restriction map for continuous functions $u(\cdot,\cdot)\mapsto u(\cdot,0)$.
To me, it seems that the natural generalisation to make is
$$(\mathcal{R}u)(\phi):=\lim_{\epsilon\rightarrow 0}u(\phi\otimes\eta_\epsilon) $$
for $u$ such that the above limit exists for each $\phi\in \mathcal{C}_c^\infty(\mathbb{R}^n)$ where $\eta_\epsilon$ is a fixed approximate identity.
Is this the sensible construction to make? And is there a convenient classification of the distributions $u$ that satisfy this property?
(This question has arisen in me trying to understand how one can in general define the analogue of the heat kernel for linear differential operators in $(x,t)$ of the form $P=(\partial_t-L_x)$ and prove the equivalence of such a solution with the notion of a fundamental solution $Pu=\delta(x)\delta(t)$. A solution kernel for such an operator should be a distribution $p\in \mathcal{D}'(\mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R})$ that in a sense restricts to a distribution on $\mathbb{R}^n\times \{y\}\times \{t\}$ which converges to $\delta_y$ in the sense of distributions as $t\rightarrow 0$.)