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Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential operators?

I would appreciate any good examples, as well as some historical outlines on the topic's development. (Shubin's classical book spends a few lines on history and motivation in the preface, but no "natural" examples. I am not aware of any historical outlines in the literature.)

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They are quite natural, in a way: once you know that usual differential operators correspond, under Fourier transform, to multiplication by polynomials, any undergrad worth her salt should ask what happens if you replace polynomials with more general functions! –  Mariano Suárez-Alvarez Mar 21 '11 at 18:30
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Yes, but they are so promiment, I expect there cases were classical differtial operator theory simply get stuck, whereas a pdo approach provides something one would actually like to have. (which is subjective, of course.) –  shuhalo Mar 21 '11 at 18:44
    
People have told me they have something to do with "d-modules" and in particular "algebraic analysis," see en.wikipedia.org/wiki/Algebraic_analysis and the corresponding links. I'm not sure how exactly algebraic analysis influenced microlocal analysis and psido's, but you might find it interesting. I think that solving the heat equation on the circle using fourier series is perhaps the first "glimpse" of psido's that I can imagine, but this is not a particularly historically motivated remark. –  Otis Chodosh May 22 '12 at 2:39
    
doesn't the Dirac operator as "square root of the laplacian" constitute one of the first examples? (although this would be a motivation coming from physics) –  Camilo Sarmiento Apr 9 '13 at 21:38

8 Answers 8

imho, this topic will be incomplete without the mention of V. Maslov. There is an original presentation of a "symbolic calculus" which he coined algebra with µ-structure in his book "Operationnal methods". I the vol. 7 of Dieudonne "treatise on analysis", there is a chapter entitled "Operators of Lax-maslov" which is coined for Fourier intergal operators if I am not mistaken. I think that WKB-method (WKB stands for Wentzel, Brillouin and Kramers) of solution of PDE is also important to cite in this context and I found the introduction of Duistermaat book, "Fourier integral operators" very interesting.

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I suggest this introduction of M.W. Wong, see 1.

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Dear @Marcelo: Adding an explicit bibliographic reference would probably be helpful in case the links rots or is otherwise not accessible. –  Ricardo Andrade Sep 15 '13 at 13:36

Let us follow the history of classical PDE with a few (hopefully) well-chosen examples illustrating the rôle of pseudodifferential operators in classical analysis.

(1) Before 1950. Prehistory. A long tradition of the russian school of Mikhlin introduced singular integrals, to be developed considerably by Calderon and Zygmund. In the late fifties and early sixties, it is quite clear that approximate inverses of elliptic operators are pseudodifferential operators and that it is a good way to prove that elliptic operators are hypoelliptic in the primitive sense that singsupp$u=$singsupp$Pu$ for $P$ elliptic.

(2) 1959. The true beginning of pseudodifferential methods in PDE: Calderon's proof in 1959 of Cauchy uniqueness for a large class of principal type operators, using a pseudodifferential factorization to prove a Carleman estimate. The first resolution of a classical analysis problem by a microlocal method.

(3) 1968. After R.T. Seeley proved the invariance of classical pseudodifferential operators by diffeomorphism, M. Atiyah and I. Singer prove the index theorem for elliptic operators.

(4) 1971. Microellipticity: introduction in 1971 by Sato and then Hormander of the wave-front-set, proof that $WF u= WF Pu$ for $P$ elliptic and more generally $WF u\subset WF Pu\cup char P$ (elliptic microlocal regularity).

(5) 1971, the apex. Proof by Sato and Hormander of the Huygens principle, formulated in the seventieth century. Although the final proof will involve Fourier integral operators, it is possible to prove the propagation of singularity theorem by a multiplier method, and for a real principal type operator $P$ and a distribution $u$ such that $Pu\in C^\infty$, $WF u$ is invariant by the flow of $H_p$, exactly as predicted by Huygens who lacked correct definitions.

(6) 1973: Proof by R. Beals and C. Fefferman of local solvability of principal type differential operators satisfying Nirenberg-Treves condition (P). A problem of local analysis solved by the introduction of nonhomegeneous class of pseudodifferential operators.

(7) 1978: subelliptic estimates. Characterization by Egorov, Hormander of operators $P$ of order $m$ such that $Pu\in H^s$ implies $u\in H^{s+m-\frac{k}{k+1}}$. The case $k=0$ is the elliptic case and the cases $k\ge 1$ involve iterated Poisson brackets of the real and imaginary part of the principal symbol of $P$.

(8) 1981: paradifferential calculus. Introduction by J.-M. Bony of pseudodifferential operators with limited regularity, proof of propagation of weak singularities for classes of nonlinear PDE.

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A slightly different motivation for fourier integral operators and pseudo-differential operators is given in the first chapter of this book. In this chapter, V.Guillemin presents this subject from the conormal bundles point of view and then shows how pseudo-differential operators are a special case of operators coming out of the Fourier distributions.

Section 3 of this chapter also provides a historic outline of this theory.

Interesting, Guillemin also shows here that pseudo-differential operators can be used to fine tune the theory of conormal distributions themselves -- giving a natural but unexpected application of pseudo-differential operators.

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My vague memories of this:

As others have mentioned, pseudodifferential operators arose from trying to establish existence and regularity theorems for a variable coefficient linear PDE using Fourier transform, extending known results for constant coefficient PDE's. I believe the earliest work in this direction, mainly for hyperbolic PDE's, is by Leray and Petrovskii. Another important early paper is by Lax, "Asymptotic solutions of oscillatory initial value problems", DMJ 1957. The first paper that explicitly defines a pseudodifferential operator was, I believe, the paper of Kohn and Nirenberg in CPAM 1965. Unlike the earlier work, I believe their focus was on elliptic PDE's.

The extension to and use on manifolds was, I believe, done by Atiyah and Singer in their original work on the index theorem, as well as Seeley.

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The same volume of CPAM 1965 also contains a paper of Hörmander that defines pseudodifferential operators (in a manifestly coordinate invariant way). –  timur Jun 1 '11 at 3:10
    
timur, thanks for the additional info. –  Deane Yang May 22 '12 at 8:50

Let me put it this way. Many natural objects in PDE theory are pseudodifferential operators. Just a few examples (besides, obviously, differential operators):

1) singular integral operators in the sense of harmonic analysis;

2) the spectral measure projectors associated to a s.a. constant coefficient differential operator, and hence all functions of that operator, in the sense of spectral theory;

3) the inverse of an elliptic operator;

4) solution operators to wave, Schrodinger, heat, younameit evolution equations.

And this list is quite incomplete. Now, pseudodifferential calculus is essentially a framework which shows the underlying common structure of all the previous examples, unifies their properties, and shows that many computations from different theories are just special cases of general theorems. To accommodate more and more interesting examples, the theory has been tweaked and enlarged several times, while keeping the same abstract structure. Thus there does not exist a single calculus, but several calculi following similar guidelines.

In addition, the procedure associating a symbol to an operator, which is at the heart of pseudodifferential calculus, is very close to the procedure of quantization in physics, and I understand it can be used to give a mathematical framework to that procedure (but I'd better leave further comments on this to the experts).

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I don't know the history at all, but I have to imagine that the language was invented to provide a context for talking about solution operators for differential equations. Consider, for example, the PDE $D f = f_0$ where $D$ is a nice differential operator. Taking Fourier transforms, this says that $P(\xi)\hat{f} = \hat{f}_0$, where $P$ is the principal symbol of $D$ (a polynomial). Everyone in the world just wants to write $\hat{f} = \frac{1}{P(\xi)}\hat{f}_0$ and take inverse Fourier transforms. In other words, solving the PDE is the same thing as finding an operator $S$ whose Fourier multiplier is $\frac{1}{P(\xi)}$. This most likely fails to be a polynomial, so $S$ is evidently not a differential operator. As far as I can tell, many of those big fat books on pseudodifferential operator theory are all about how to invert as many operators as possible in this sense while salvaging as much regularity as you can. It gets extremely subtle, but I think the motivation is fairly close to the surface.

Aside from that, you might also be led to invent pseudodifferential operators if you cared deeply about the spectral theory of differential operators. The spectral theorem for an operator $T$ is more or less equivalent to the existence of a "functional calculus", i.e. a sensible way to form operators $f(T)$ out of various classes of functions $f$ on the spectrum of $T$. For differential operators (especially on non-compact domains where there need not be a nice eigenspace decomposition), the functional calculus is often obtained via the Fourier transform, and the pseudodifferential calculus manifests itself.

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Many applications of pseudo-differential operators, especially to boundary value problems for elliptic and hyperbolic equations, can be found in the book by F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vols 1 and 2, Plenum Press, New York, 1982.

A different kind of pseudo-differential operators, typically with non-smooth (e.g. homogeneous) symbols, appears in probabilistic applications - such operators emerge as generators of Markov processes with jumps. See

S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkh\"auser, Basel, 2004,

and

N. Jacob, Pseudo-differential operators and Markov processes, Vols 1-3, Imperial College Press, London, 2001-2005.

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