# A text about Schwartz distributions in vector bundles

If $$M$$ is a smooth manifold, one may talk about the space of test functions $$\mathcal D (M)$$ and its topological dual $$\mathcal D ' (M)$$ - the space of Schwartz distributions on $$M$$.

Now, if $$E \to M$$ is a vector bundle (of finite rank), it is intuitively clear that one should be able to talk about the space of "test sections" $$\mathcal D (M; E)$$ and its topological dual $$\mathcal D ' (M; E^*)$$, the space of "$$E^*$$-valued distributions on $$M$$". One should then be able to talk about distributional solutions to PDEs in vector bundles, about hypoellipticity, about Dirac distributions in bundles etc.

I believe that this extension of distribution theory has been written, but I do not know where to look for it, be it in the form of a monograph, book chapter or article. Could anyone please help me with a (canonical, if possible) reference? Thank you.

• I see that there are some very nice books about distributions on $\mathbb R^n$ with values in some vector space $V$. One would only have to glue together these distributions in order to obtain what I am looking for - so maybe it has already been done. Aug 26, 2022 at 9:50

This is pretty standard material and I don't know of a specific reference that answers your question and only that. You will find lots of references if you search for "distributions on manifolds". Perhaps the emphasis will be on scalar distributions, not vector valued ones, but the differences are minor, once you make sure all the partitions of unity used in the theory are adapted both to an atlas of coordinates and to a trivialization of the vector bundle.

A classic reference is the book of de Rham (he only deals with scalars, forms and polyvectors):

de Rham, Georges, Differentiable manifolds. Forms, currents, harmonic forms. Transl. from the (1955) French edition by F. R. Smith. Introduction to the English ed. by S. S. Chern, Grundlehren der Mathematischen Wissenschaften, 266. Berlin etc.: Springer-Verlag. X, 167 p. (1984). ZBL0534.58003.

Nowadays, books mostly include a single chapter or section on the theory in the generality that they need. To see specifically distributional sections of vector bundles, see for instance:

§1.1 in Bär, Christian; Ginoux, Nicolas; Pfäffle, Frank, Wave equations on Lorentzian manifolds and quantization., ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society Publishing House (ISBN 978-3-03719-037-1/pbk). viii, 194 p. (2007). ZBL1118.58016.

§1.1 in Tarkhanov, N. N., Complexes of differential operators. Rev. a. upd. transl. from the Russian by P. M. Gauthier, Mathematics and its Applications (Dordrecht). 340. Dordrecht: Kluwer Academic Publishers. xviii, 396 p. (1995). ZBL0852.58076.

Update: The question about a reference for a Schwartz Kernel Theorem on manifolds is more delicate. In my understanding, it does follow from the general results on tensor products of nuclear locally convex vector spaces originally proven by Grothendieck. But certainly, in order not to get into the weeds of functional analysis every time you'd like to refer to that result, it's nice to have a handy reference. I've found a version of it in Thm.1.5.1 of Tarkhanov's book referenced above. Taking a hint from wikipedia, a more classical reference (though it covers only scalar test functions and distributions) may be Thm.23.9.2 of

Dieudonné, J., Treatise on analysis. Volume VII. Translated by Laura Fainsilber, Pure and Applied Mathematics (New York) 10-VII. Boston, MA etc.: Academic Press, Inc. xiv, 366 p. {\$} 69.95 (1988). ZBL0672.58044.

• I agree with you that most of the theory is about glueing together local data. The thing is that I might want to use, say, Schwartz's kernel theorem for operators between bundles without having to construct first the whole theory myself from first principles. Aug 26, 2022 at 13:29
• @AlexM. I've added references to a sufficiently general version of the Schwartz Kernel Theorem. Making your question specifically about that, I think, makes it more interesting, as the desired reference should go beyond the basic definitions. I would go so far as recommend editing your original question and changing its title, to reflect what you are actually interested in. Aug 26, 2022 at 15:59
• The kernel theorem was just a random, even though notable, example. I am interested in having analogues in bundles of most theorems in distribution theory, readily available for use in publications, without me having to eat up supplementary space in the journal in order to derive them myself. Aug 26, 2022 at 16:46
• @AlexM. Fair enough, please try to make a list of such theorems. Aug 26, 2022 at 17:24