What is the "dual" of the space of currents?

Question

On a smooth maniflod $M$ of dimension $n$, a current of degree $n-p$ is a functional on the space of compactly supported differential $p$-forms which is continuos. We denote the space of currents of degree $n-p$ by $D^{'n-p}(M)$. If we consider the functionals on $D^{'n-p}(M)$ with an approperiate comapctness and continuity assumptions, then what are these functionals? Are they just differential $p$-forms, or could be more then that?

I am asking this question because I want to understand why there is no definition for the pullback of current $T$ in general (as far as I know). Let $f: M_1 \to M_2$ be a map between manifolds. Then the pullback $f^*T$ should be defined as (formally) $$\langle f^*T, u\rangle = \langle T, f_*u \rangle.$$ Here $u$ is a differential $p$-form, in particular, $f_*u$ is well-defined as a current under some compactness assumption. Thus the problem is to make sense of $\langle T, f_*u \rangle$, i.e. can a current be a functional on the space of current? (Of course, it is enough to have $T$ be a functional on the space $\{f_*u\}$, that is why pullback of current is well-defined for submersion maps).

Answer 1

This is a question of functional analysis: Start with a locally convex space $X$, form its dual $X^*$ consisting of continuous linear functionals on $X$. Now $X^*$ has several locally convex topologies $\tau$ with the following property:

For any $x\in X$ the linear map $L_x:X^*\to\mathbb{R}$ is $\tau$-continuous.

For such topology $\tau$ we form the bidual $X^{**}_\tau$ consisting of linear functionals on $X^*$ that are $\tau$ continuous. From the choice of $\tau$ we deduce that we have a natural map

$$X\ni x\mapsto L_x\in X^{**}_\tau. $$

The question is if there exist topologies $\tau$ on $X^*$ such that the above map is an isomorphism of locally convex spaces.

A natural topology on $X^*$ is the smallest locally convex topology such that all the linear functionals $L_x$ are continuous. Let's call this $\tau_0$. (Traditionally it is denoted by $\sigma(X^*, X)$.)

We can ask a more refined question: for which locally convex spaces the bidual $X^{**}_{\tau_0}$ coincides with $X$.

There is know a large class of such spaces namely the nuclear spaces. For details see

Gelfand & Shilov: Generalized Functions., vol 2.

For example the space $X=C^\infty(M)$, $M$ compact has this property.

Answer 2

This follows the setup of [De Rham: Differentiable manifold, Springer-Verlag 1984]. Smooth differential form with compact support form the space $\Gamma_c(\Lambda^pT^*M) = \Gamma_{C^\infty_c}(\Lambda^pT^*M)$. Its dual space is the space of distributional sections $\Gamma_{\mathcal D'}(\Lambda^pTM)$. If the manifold is orientable, smooth $(n-p)$-forms can be embedded into $\Gamma_{\mathcal D'}(\Lambda^pTM)$ by using the action $\int_M \phi\wedge\psi$. If the manifold is not orientable, passing to the orientable double cover and considering forms in the $\pm1$ eigenspaces of the pullback with the deck-transformations (these correspond to "forme pair ou impair" of De Rham) one can carry this over. Going to the completion, we can also view currents as distributional sections $\Gamma_{\mathcal D'}(\Lambda^{n-p}T^*M$.

For the pullback: you can pullback the bundle $\Lambda^{np}T^*M$, but the distributional coefficients (in a local frame) only under a diffeomorphism, or under a smooth mapping where the image of the tangent mapping contains the wave front set of each distributional coefficient.

In other words: pulling back the bundle is not the obstruction; pulling the distributional coefficients is it!