How to show that a Schwartz distribution is in a Lebesgue or Sobolev space?

Question

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.)

Answer 1

As has been mentioned in a comment your question is very broad. One possible answer is to use the space of tempered distributions $\mathcal{S}'(\mathbb{R}^n) \subset \mathcal{D}'(\mathbb{R}^n)$, it consists of the linear functional continuous with respect to $\mathcal{S}(\mathbb{R}^n)$ which is the Schwartz space. In other words it can be shown that the Fourier operator $\mathcal{F}: \mathcal{S}'(\mathbb{R}^n) \longrightarrow \mathcal{S}'(\mathbb{R}^n)$ is a continuous (in the sense of distributions) isomorphism, extension of Fourier operator $\mathcal{F}: L^1(\mathbb{R}^n) \longrightarrow L^\infty(\mathbb{R}^n)$, and using $\mathcal{F}^{-1} : \mathcal{S}'(\mathbb{R}^n) \longrightarrow \mathcal{S}'(\mathbb{R}^n)$ "you can go back". This probably can notice slightly in the spaces of Hilbert-Sobolev $H^s(\mathbb{R}^n)$, for example it can be shown that $\mathcal{E}'(\mathbb{R}^n) \subset \bigcup_{s \in \mathbb{R}} H^s(\mathbb{R}^n)$, where $\mathcal{E}'(\mathbb{R}^n)$ is the space of distributions with compact support, dual space of regular functions. I think that this is possible technique.