# Distributions on manifolds

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.

• Is this something you are looking for? – Ilya Jul 12 '13 at 12:26
• @Ilya: Sorry, it's not, at least I see no connection at all. I think it's just a coincidence of names. I'm speaking of distributions as in generalized functions, see my link in the question. An example of distribution on $S^1$ would be a linear functional which, evaluated in any infinitely differentiable function on the unit circle, would return its value in a particular point. Or the integral of the $C^\infty$ function over the circle. Or individual Fourier coefficients. – The Vee Jul 12 '13 at 12:59

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

• Thank you! Your course notes are very good, very thorough, well-structured and extremely inspiring! They did not answer my question in full but they contain all the information I need at this moment for the work I mentioned. I have a suspicion that the dual of $C^\infty(S_1)$ is trivially isomorphic to the space of all $2\pi$-periodic distributions on $\mathbb{R}$; it also holds in the latter case that a Fourier series converges in $\mathcal{D}'(\mathbb{R})$ if the coefficients are bounded by a polynomial in n. Could you please confirm this? – The Vee Jul 12 '13 at 17:05
• Yes, your suspicion is completely justified: the dual of $C^\infty(S^1)$ (the distributions on the circle) is identical with the periodic distributions on the line. And, yes, the Fourier series of such things converge exactly when the coefficients have at worst polynomial growth. – paul garrett Jul 12 '13 at 17:11
• @paulgarrett; Indeed, your course notes are very motivating and interesting; thanks; – Inquisitive Dec 16 '14 at 13:06

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.)