Let $\phi \in \mathcal{S}'(\mathbb{R}^n)$, the space of tempered distributions. How can one restrict $\phi$ to the hyperplane $x_1 = c$ for some $c \in\mathbb{R}$?
The answer to this question suggests that this is possible if the distribution is not too singular and that this restriction would mean $\phi$ can be viewed as the map $$\mathbb{R}\rightarrow S'(\mathbb{R}^{d-1}), \quad t\mapsto \phi(t, x_2, \ldots, x_n).$$ What does not being too singular mean in this context and how does one define the new distribution $\phi(t, x_2,\ldots,x_n) \in \mathcal{S}'(\mathbb{R}^{n-1})$?
I would think one can define such a restriction by letting $\phi$ act on the following subset of $\mathcal{S}(\mathbb{R}^{n-1})$: $$\{ f \in \mathcal{S}(\mathbb{R}^{n-1}) : f = g(c, x_2, \ldots, x_n) \text{ for } g \in \mathcal{S}(\mathbb{R}^{n})\}.$$ But it seems undesirable to let the restricted $\phi$ act on only a subset of $\mathcal{S}(\mathbb{R}^{n-1})$.
