How to show that a Schwartz distribution is in a Lebesgue or Sobolev space? It is known that all $L^p$ spaces (and, consequently, all $W^{s,p}$ spaces) can be embedded in the space of Schwartz distributions $\mathcal D '$. There is a problem, though: how do I check whether some given distribution $u \in \mathcal D '$ belongs to any of the Lebesgue or Sobolev spaces mentioned above? (There are problems requiring the student to show this, and I have no clue what technique(s) to use and how to approach them.)

Although hard to answer without specific examples in mind, this is a good question. The way I would approach this kind of problems is as follows. Let $\rho$ be a compactly supported smooth function on $\mathbb{R}^d$ with integral equal to $1$. Pick your favorite contant $L>1$. For $r\in\mathbb{Z}$, define $\rho_r(x)=L^{-rd}\rho(L^{-r}x)$ and let $u_r=\rho_r\ast u$. Namely, this is the smooth function
$$
u_r(x)=\langle u(y),\rho_r(x-y)\rangle_y
$$
which is a better way of writing the distributional pairing while retaining the expressive power of the merely formal
$$
\int_{\mathbb{R}^d}u(y)\rho_r(x-y)\ d^dy\ .
$$
Note that $\lim_{r\rightarrow -\infty}u_r=u$ in $\mathcal{D}'$ with its correct topology, i.e., the strong topology.
The first thing to do is check that $u_r$ is in $L^p(\mathbb{R}^d)$.
Then I would look for a bound of the form
$$
||u_r-u_{r-1}||_{L^p}\le K L^{\epsilon r}
$$
for some constants $K\ge 0$ and $\epsilon>0$.
If the bound holds, then $u_r$, $r\rightarrow -\infty$, will converge in $L^p$ to $u$ seen as an element of that subspace of $\mathcal{D}'$.
