Distributions as derivatives

Question

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?

Answer 1

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.

Answer 2

This is perhaps tangential to your query but I am posting it in the hope that it might contain useful information. I will start with the case of distributions on a compact interval, which we can assume to be $[0,1]$. Then, as you point out, every distribution thereon is a (repeated) derivative of a continuous function but this representation is not unique. However, the level of non-uniqueness is rather mild, in fact, if we have two such representations as $D^n F=D^m G$, then we must have that $I^m F-I^n G$ is a polynomial of degree at most $m+n-1$, where $I$ is the operator which assigns to a continuous function on $[0,1]$ that primitive which vanishes at $0$. This simple fact was used by J. Sebastiao e Silva to develop an elementary approach to the theory of distributions which, I believe, answers in part your query. The basis of his approach is the following system of axioms:

1. The space $C[0,1]$ can be embedded into a vector space $C^{-\infty}[0,1]$ on which there is defined a linear operator $D$ which coincides with differentiation on the continuously differentiable functions;

2. For each $f \in C^{-\infty}[0,1]$, there are a continuous $F$ and a natural number $n$ so that $f=D^n F$;

3. If $F$ is a distribution with $D^n F=0$, then $F$ is a polynomial of degree at most $n-1$.

The distributions of Schwartz satisfy these conditions of course but Sebastiao e Silva showed that this system is categorical (i.e., any two models coincide in the natural sense) and secondly that one can show directly that it has a model, using very simple facts at the level of an introductory calculus case. He then gave a systematic introduction to the theory of distributions at this level, avoiding the use of duality theory for locally convex spaces. The one-dimensional theory can then be extended to the multivariate one using standard methods as can the case of distributions on open, rather than compact, sets. (I can supply references, if so desired).

Returning to your original query, at least partial answers in elementary terms can be given to your original request.