Distributions as derivatives Distributions can be viewed as derivatives of continuous functions, see Rudin's book on Functional Analysis. This representation has several drawbacks:


*

*One cannot read off the order of a distribution,

*nor the support,

*the representation is not unique.
More generally, one can say that every distribution (also on manifolds and vector bundles thereon) can be written as a sum of derivatives of Radon measures. 
My question is: can this representation be made unique, by, say, insisting that each summand be of minimal order? In that case, can one read off the support as being equal to the support of the involved measures? Likewise for the order as being the maximal order of derivation?
 A: Maybe more a lengthy comment than an answer. Let me work with temperate distributions (dual of the Schwartz space $\mathscr S(\mathbb R^d)$). Let $u\in  \mathscr S'(\mathbb R^d)$ and let us define the Fourier multiplier
$$
\langle D\rangle=(1+\vert D\vert ^2)^{1/2},\quad \text{so that Fourier$(\langle D\rangle u)=\langle \xi\rangle \hat u.$}
$$
We have for any $s\in \mathbb R$,
$
u=\langle D\rangle^s\langle D\rangle^{-s} u.
$
If $u$ happens to be in $H^{-\infty}=\cup_{s\in \mathbb R}H^s$, for some $s$, we get 
$\langle D\rangle^{-s} u \in H^t(\mathbb R^d)$ with $t>d/2$ and thus is a continuous function: take $s_0$ to be the infimum  of the $s$ for which that property holds and take $s_1>s_0$: we find
$$u=\langle D\rangle^{s_1}\underbrace{\langle D\rangle^{-s_1} u}_{\text{continuous function}},$$
so $u$ appears as some sort of normalized derivative of a continuous function. There are variations on this topic with powers of the harmonic oscillator
$\mathcal H=-\Delta+\vert x\vert^2$ replacing powers of $\langle D\rangle$ or more generally powers of some globally "elliptic" operator with an "explicit" inverse. The representation is heavily dependent on the choice of the elliptic operator, but when that choice is made, can be normalized, e.g. thanks to the existence of a $s_0$ in the example above.
