In the terminology of Gel'fand, Generalized Functions Volume 1, a regular distribution, $\Lambda_f$, is one that is obtained from a locally integrable function, $f$, by integration of the function times a test function:
$$\Lambda_f(\phi)=\int_{\mathbb{R}^n}\!f(x)\phi(x)\,dx\qquad(\phi\in C^\infty_c(\mathbb{R}^n)).$$
I know that the Dirac delta distribution, defined by
$$\delta_p(\phi)=\phi(p)\qquad(\phi\in C^\infty_c(\mathbb{R}^n),\,p\in\mathbb{R}^n)$$
is not a regular distribution. For the $n=1$ case, there is a proof in Gel'fand on page 4 of Volume 1, there is this post: How to prove that the Dirac delta is not a function?, and it is not hard to prove using the $C^\infty$-Urysohn Lemma, even for general integer $n$. (Find a mollified sequence of $C^\infty_c(\mathbb{R}^n)$ functions converging pointwise to the characteristic function of $\{p\}$, use Dominated Convergence and the fact that the Lebesgue measure of $\{p\}$ is zero, to get a contradiction.)
I believe that it must be true as well for all derivatives of the delta distribution. Again, for the $n=1$ case, there is this website from Professor Tao: https://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/ in which one finds:
Exercise 11 (Derivative of the delta function) Let
. Show that the functional
for all test functions
is a distribution which does not arise from either a locally integrable function or a Radon measure. (Note how it is important here that
I have been trying to prove that $\delta_p'$ is not regular in $\mathbb{R}^1$; that is, that there does not exist $f\in L^1_{\rm loc}(\mathbb{R})$ such that $\delta_p'=\Lambda_f$, hoping that such a proof would be generalizable to $\mathbb{R}^n$. However, I haven't had any luck even with $n=1$. So my simplified first question, setting $p=0$ and $n=1$, is "why can't $\delta'$ be $\Lambda_f$ for some $f\in L^1_{\rm loc}(\mathbb{R})$?" If that can be answered, then the next question is can it be extended to $\mathbb{R}^n$? Is it true that, given any multi-index $\alpha$, there is no $f\in L^1_{\rm loc}(\mathbb{R}^n)$ such that $D^\alpha\delta_p=\Lambda_f$?
In working on the $n=1$, $p=0$ case, I start with the obvious:
\begin{equation}\tag{1}
-\phi'(0)=-\delta(\phi')=\delta'(\phi)=\Lambda_f(\phi)=\int_{-\infty}^\infty\!f(x)\phi(x)\,dx
\qquad(\phi\in C^\infty_c(\mathbb{R})),
\end{equation}
and look for a sequence of test functions that generate a contradiction.
The trick of letting $\{\phi_k\}$ be a mollified sequence in $C^\infty_c(\mathbb{R})$ such that $\phi_k$
converges pointwise to $\chi_{\{0\}}$ doesn't seem to help because it appears that $\phi'_k(0)$ will
likely have to approach $0$ and therefore not generate a contradiction. Letting $\{\phi_k\}$ just
be a bump that narrows down to $0$ but whose height remains constant, also doesn't seem to work, since again, the derivatives will be $0$ at the origin. I also hoped I might be able to make use of the fact that
if $\phi_1$ is a fixed element of $C^\infty_c(\mathbb{R})$ such that
$$\int_{-\infty}^\infty\!\phi_1(x)\,dx=1,$$
then any $\phi\in C^\infty_c(\mathbb{R})$ can be expressed as
\begin{equation}\tag{2}
\phi=\phi_0+\Biggl[\int_{-\infty}^\infty\!\phi(x)\,dx\Biggr]\phi_1,
\end{equation}
where $\phi_0\in C^\infty_c(\mathbb{R})$ and is the derivative of a function in
$C^\infty_c(\mathbb{R})$,
as used by Gel'fand on page 40 of Volume 1; however, I couldn't find a way where that would help.
Best Answer
If $\delta'$ was in $L^1_{\mathrm{loc}}$, then it would have a continuous primitive which would have the properties of the Dirac delta, which is a contradiction.
You can iterate this reasoning $n$ times to get the same result for the higher derivative $\delta^{(n)}$.