Distributions on manifolds

Question

Wikipedia entry on distributions contains a seemingly innocent sentence

With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb{R}^n$ by any (paracompact) smooth manifold.

without any reference cited. I went through Vladimirov, Demidov, Gel'fand & Shilov but could not find a single mention of the latter concept. Of course, I have an intuitive feeling of how to go about this, but I would need to use generalized functions on $S^1$ in my work and I don't want to inefficiently re-discover the whole theory if it exists anywhere already. Could anyone point me at a reference where I could learn more about distributions on smooth manifolds?

NB this is not the same question as distributions supported, or concentrated, on a manifold embedded in $\mathbb{R}^n$. My space of test functions would also be defined on the same manifold so transverse derivatives are not defined.

Answer 1

The case of a circle or products of circles is much nicer than the general case of manifolds, since there is a canonical invariant ("Haar") measure! Further, circles are abelian, compact Lie groups. And connected.

Thus, smooth functions are identifiable as Fourier expansions with rapidly decreasing coefficients, and distributions have Fourier expansions with at-most-polynomially-growing coefficients.

There is a useful gradation in between, by Levi-Sobolev spaces, etc.

One version of this is in my course notes http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/04_blevi_sobolev.pdf

Answer 2

There is some material distributed on the four volumes of "The Analysis of Linear Partial Differential Operators" by Lars Hörmander, where the very basic definitions can be found in section 6.3 "Distributions on a Manifold" of the first volume. (I am not entirely happy as there might be literature dealing specifically with this subject.)