In Stein's Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, he defines tempered distribution ($\mathscr S'$) as continuous linear functionals from the Schwartz class. Here, the continuity is given with respect to the family of seminorms
\begin{equation*}\|\Phi\|_{\alpha,\beta}=\sup_{x\in\mathbb R^N}|x^\alpha\partial^\beta_x\Phi(x)|.\end{equation*} Then, without any further clarification, he proceeds to discuss convolutions between a tempered distribution and functions in the Schwartz family. This is where I get puzzled. If the tempered distributions are functionals, what is this convolution supposed to mean? It seems as if he (and every other source, for that matter) was assuming these functionals can be clearly identified with some functions. My question is: how do you make this identification?
I was thinking, as Schwartz functions are in $L^2$, maybe the identification is the one given by Riesz representation theorem. However, I think this is not possible as the topology we are considering in the Schwartz class is different from that of $L^2$. Moreover, while discussing $H^p$ spaces, he claims that, for $p>1$, $L^p$ is the same as $H^p$. Here, he is using again this identification that I don't quite get and, if my hypothesis was correct that the identification is made through Riesz representation theorem, this should mean $H^2=L^2=\mathscr S'$. This seems a bit strange to me. Another thing that's worrying me as well is the fact that he is discussing, without a prior definition, bounded distributions. Of course, if these were elements of $L^2$'s dual, they would be automatically bounded, so this is another hint that my original assumption about the identification is wrong.
I think this is a very basic question, but I don't find any source in which this is specifically discussed and clarify. How can we talk about an object as both a tempered distribution (an element of $\mathscr S'$) and a function defined on $\mathbb R^N$?
Best Answer
We don’t need to identify distributions with functions to meaningfully discuss convolution with distributions. Most distributions can’t be identified with functions. Any operation normally defined between functions we define for distributions using duality, by determining how it acts on Schwartz functions. Let $\langle g,\phi\rangle$ be the evaluation pairing between a distribution $g$ and a Schwartz function $\phi$, i.e. $g: \phi \mapsto \langle g,\phi\rangle$. Notice that if $g$ is a locally integrable function, this pairing is just the $L^2$ inner product.
The strategy for determining the proper dual formulation for a statement about distributions is to assume all objects involved are nice functions, then move things around until we have an evaluation pairing between a distribution and a Schwartz function. For example, to make sense of the the action of the distribution $g*\psi$ on a Schwartz function $\phi$, where $g$ is a distribution and $\psi$ is Schwartz, if $g$ were nice we could use Fubini to get
$$\begin{align} \langle g*\psi, \phi \rangle &= \int \phi(x) \int g(y)\psi(x-y) dydx \\ &= \int g(y) \int \phi(x)\psi(x-y)dx dy \\ &= \langle g, \phi*R\psi\rangle \end{align}$$
Where $R$ is the reflection operator $R:\psi(x)\mapsto \psi(-x)$. With this in mind, we define by duality $\langle g*\psi, \phi \rangle = \langle g, \phi*R\psi\rangle$. This determines the behavior of the distribution $g*\psi$ uniquely, since we know how $g$ acts on Schwartz functions. Of course, there’s some details here you can check, like the fact that $g*\psi$ is indeed a tempered distribution, that convolution of Schwartz functions is Schwartz, etc.
Another example: to define derivatives of a distribution, we determine its action on smooth functions by pretending the distribution is smooth and integrating by parts until we have something meaningful:
$$ \langle \partial^\alpha g,\phi\rangle = (-1)^{|\alpha|} \langle g, \partial^\alpha \phi\rangle$$
This equality ultimately defines $\partial^\alpha g$.
Once you’ve established these duality-based definitions, you can informally treat distributions as if they’re functions, even though to be precise you need to work with their actions on Schwartz functions. This is what’s going on behind the scenes in Stein.