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Let $M$ be a smooth manifold.

I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on $M$). Sadly the best results I could find showed at best that pseudo-differential form a sheaf of left $C^{\infty}$-modules without treating the question of whether the multiplication is well defined nevermind associative. So my question is rather simple:

Do pseudo-differential operators form a sheaf of (associative?) $\mathbb{C}$-algebras on $M$? If not, what fails?

I'm pretty sure that if one considers pseudo-differential operators modulo smothing operators than these do form a sheaf of algebras (However a precise reference here would be welcome). As for the entire space of pseudo-differential operators this seems rather non-trivial to me (I'm actually not entirely sure whether this should be true or not). I apologize if this is too elementary for this site.

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  • $\begingroup$ My amateur appraisal on this is that there is definitely an issue about composition, especially if you want composition to be associative, if we can't take a quotient by smoothing operators. I don't have an example off-hand, but this is surely vaguely similar to the old chestnut that $(1*\delta')*H\not=1*(\delta'*H)$ where $H$ is the Heaviside function. The problem is that not-compact-support distributions (for example) don't form a reasonable convolution algebra, certainly not compatibly with a purported action on nice functions... and so on. $\endgroup$ – paul garrett Feb 19 '18 at 22:20
  • $\begingroup$ @paulgarrett Aha! So my suspicions were well founded! If it is indeed true that associativity fails it is rather unfortunate that this fact is glossed over so much in the literature. I hope some expert will eventually find this post and clear this up. $\endgroup$ – Saal Hardali Feb 19 '18 at 22:24
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Pseudodifferential operators are operators whose Schwartz kernel is smooth outside of the diagonal and is conormal with respect to the diagonal (differentiating finitely many times with respect to smooth vector fields tangent to the diagonal produces a distribution in a certain Besov space).

From this definition one immediately sees that pseudodifferential operators do form a presheaf of abelian groups, albeit in a nonobvious way: a Schwartz kernel on M×M can be restricted to U×U. But there is no way this could be a sheaf: knowing restrictions to U×U and V×V such that U and V cover M does not tell what happens on M×M outside of U×U and V×V.

Sheafification produces germ equivalence classes of conormal distributions near the diagonal.

This is a larger sheaf than equivalence class of pseudodifferential operators modulo smoothing operators (which do form a sheaf for trivial reasons). Indeed, all germs of smooth functions near the diagonal produce different elements in the sheafification, but become equal once one mods out smoothing operators.

The above talks about sheaves of abelian groups. Pseudodifferential operators cannot always be composed (and accordingly do not form presheaves of algebras); one needs conditions like properness of support.

However, the sheafification and the equivalence classes modulo smoothing operators both do form sheaves of algebras.

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  • $\begingroup$ Do you know a reference for the equivalence between the definition you give and the standard one? $\endgroup$ – Saal Hardali Feb 21 '18 at 12:50
  • $\begingroup$ @SaalHardali: The definition of a pseudodifferential operator is as standard as it could ever get: it is used by Hörmander himself in his 4-volume treatise. $\endgroup$ – Dmitri Pavlov Feb 21 '18 at 13:48
  • $\begingroup$ What i meant was the definition through fourier integral operators and symbol classes $\endgroup$ – Saal Hardali Feb 21 '18 at 13:51
  • $\begingroup$ @SaalHardali: Yes, the definition via conormal distributions is what Hörmander uses in his 4-volume treatise. The original reference is Chapter II of Hörmander's Fourier integral operators I. $\endgroup$ – Dmitri Pavlov Feb 21 '18 at 13:54
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See the following papers, where it is shown that suitable spaces of invertible Fourier integral operators form an infinite dimensional Frechet Lie group with corresponding spaces of pseudifferential operators as Lie algebra. There are also papers by Omori, Maeda, etc with somewhat weaker results.

  • MR0863703 Adams, Malcolm; Ratiu, Tudor; Schmid, Rudolf A Lie group structure for Fourier integral operators. Math. Ann. 276 (1986), no. 1, 19–41. (Reviewer: J. J. Duistermaat) 58G15 (22E65 58B25)

  • MR0826458 Adams, Malcolm; Ratiu, Tudor; Schmid, Rudolf A Lie group structure for pseudodifferential operators. Math. Ann. 273 (1986), no. 4, 529–551. (Reviewer: J. J. Duistermaat) 58G15 (22E65 35S05 47G05 58B25)

  • MR0823314 Adams, Malcolm; Ratiu, Tudor; Schmid, Rudolf The Lie group structure of diffeomorphism groups and invertible Fourier integral operators, with applications. Infinite-dimensional groups with applications (Berkeley, Calif., 1984), 1–69, Math. Sci. Res. Inst. Publ., 4, Springer, New York, 1985. (Reviewer: J. J. Duistermaat) 58D05 (22E65 58B25 58F99 58G15)

  • MR0815694 Schmid, Rudolf; Adams, Malcolm; Ratiu, Tudor The group of Fourier integral operators as symmetry group. XIIIth international colloquium on group theoretical methods in physics (College Park, Md., 1984), 246–249, World Sci. Publishing, Singapore, 1984. 58G15 (22E65)

Added:

Pseudodifferential operators are not local operators in general. So it does not make sense to speak of sheaf properties in a naive sense. I have no idea what sheafification would lead to; I doubt, that it is anything useful or interesting. Microlocal analysis does look superficially like that, but not really.

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  • $\begingroup$ Interesting! Does this imply that they form a sheaf? (perhaps of lie algebras by the looks of it). $\endgroup$ – Saal Hardali Feb 19 '18 at 22:35

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