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.)
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.
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 seventeenth 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.
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 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 gives a mathematical framework for the quantization procedure in physics.
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.
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.
A slightly different motivation for fourier integral operators and pseudo-differential operators is given in the first chapter of this book - Fourier Integral Operators, chapter V. W. Guillemin: 25 Years of Fourier Integral Operators. 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.
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 $\mu$-structure in his book "Operationnal methods". In the vol. 7 of Dieudonne "treatise on analysis", there is a chapter entitled "Operators of Lax-Maslov" which is coined for Fourier integral 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.
Disclaimer. This answer stems from a draft of an entry on the symbol of a singular integral operator I am writing for the Wikipedia, thus it is by no means complete. From the sources I have read until now (for example Gaetano Fichera's papers [1], p. 475 and [2], pp. 52-54 and Vladimir Maz'ya's book [5], p. 143), it seems that the motivation for the development of the theory of pseudodifferential operators lies in the development of the theory of singular integral operators, emerged as means for solving variable coefficients PDEs: precisely, Fichera and Maz'ya identify the starting point of the theory with the discover of the symbol of a singular integral operator made by Solomon Mikhlin.
Short answer
While studying the $3$-dimensional potential generated by a plane thin disk (of arbitrary shape), Francesco Tricomi succeeded for the firs time ever in giving a closed form expression for the composition of two $2$-dimensional (constant coefficient) singular integral operators (see [11], [12]). Tricomi's work led Solomon Mikhlin (see [5], pp. 2124-2125, [6], pp. 535-536 and for a brief but fairly complete historical survey see [8], §1.1.4 pp. 8-10), followed by Georges Giraud, to the discovery of the concept of the symbol of a singular integral operator. Be it noted that at that time, due to the form in which it was discovered, it was not clear that the symbol was strictly related to the Fourier transform the kernel respect to the integration variable, therefore it was not clear its relation with the Fourier transform of partial derivatives: however, its discovery made clear that the symbol is the key to uncover the algebraic structure of the composition of such operators and the structure of their spaces. Later on, Solomon Mikhlin ([2], p. 54 and [8], §1.1.7 p. 11) discovered that the symbol is the partial Fourier transform, respect to the integration variable, of the kernel of the singular integral operator. The use of the machinery of multidimensional Fourier integrals, developed by Alberto Calderón and Antoni Zygmund, became thus one of the strong points of the theory, and finally Kohn & Niremberg and Hörmader synthesized operator algebras which include both linear partial differential and singular integral operators: the theory of pseudodifferential operators was born.
A more detailed survey
Background on PDE theory at the beginning of the twentieth century: reduction to integral equations.
One of the more extensively used techniques for the solution of PDEs which was developed at the beginning of the 20th century was the "Method of Potentials": it stemmed from the methods developed the in the nineteenth century for solving Laplace's equation and consist in representing the solution of a given problem as the sum of properly chosen potential type integrals (volume, single and double layer potentials) (see [8], Chapter III, p. 39 and §18, pp. 44-49 and [9], Chapter III, p. 49 and §18, pp. 54-59). These integral representations are obtained for example by using Hadamard's elementary solutions and Levi functions (now called parametrices): the set of integral equations obtained is hopefully analyzable by applying Fredholm's theory.
However, this is not always so. These integral equations are, as a rule, often singular i.e. do not exist in the ordinary sense but only in the sense of their principal values: in this case, Fredholm's theory cannot be applied directly. This happens, for example, for the Oblique Derivative Problem for Laplace's equation, as Henri Poincaré and Gaston Bertrand noted in their analysis of the problem of tides, de facto the first oblique derivative ever posed (see for example [3], pp. 251-252). Thus the time was ready for a theory of singular integral equations to be developed.
The symbol of a singular integral: Tricomi, Mikhlin and Giraud.
Tricomi, while studying problem of determining the harmonic potential of a plane thin disk (of arbitrary shape) immersed in $\Bbb R^3$, found a formula for the explicit calculation of the composition of two $n$-dimensional (precisely $2$-dimensional) singular integral operators (see [11] and [12], §4 pp. 107-112 and also [8] §1.2, pp. 2-4). Building on the the work of Tricomi, Mikhlin discovered the concept of symbol in the form of a complex Fourier series (see Fichera [2], p. 53): let
$$
Su(z)=\iint u(\zeta)K(z,z-\zeta)\mathrm{d}\xi\mathrm{d}\eta\quad z=x+iy,\;\zeta=\xi +i\eta,\;u\in L^p\label{1}\tag{1}
$$
be a syngular integral operator whose kernel has the form
$$
K(z,\zeta)=\frac{1}{|\zeta|^2}f\left(z\frac{\zeta}{|\zeta|}\right),
$$
and let
$$
f(z,\theta)=\sum_{h\in\Bbb Z\setminus \{0\}} c_h(z)e^{ih\theta}.
$$
where the zero order term is missing since the condition
$$
\int\limits_{-\pi}^{+\pi} f(z,e^{i\theta})\mathrm{d}\theta=0
$$
must be fulfilled in order for the integral \eqref{1} to exists if, for example, $u\in\operatorname{Lip}$. Then Mikhlin defines the symbol of $S$ as
$$
\sigma(z,\zeta)=\sum_{h\in\Bbb Z\setminus \{0\}} c_h(z)\frac{-i^{|h|}}{|h|}
e^{ih\theta}.
$$
In the paper [5] he shows how to extend the results of [4] to singular integrals defined on a closed surface.
Georges Giraud, who was the leading mathematician in the development of the method of potentials (to the point that, according to Carlo Miranda ([7], p. 44 and [8], p. 54) he provided "the most general and definitive contributions"), was consequently quite interested in the analysis of singular integrals.
Closely after the appearance of the works [4] and [5], Giraud published the note [3] in the Comptes rendus, giving (without proof) formulas for the symbol of a singular integral operator, expressed in the form of a series of harmonic polynomials, and for the composition of two singular operators that extends the work of Mikhlin to the $(n\ge 2)$-dimensional case: nor Giraud's nor Mikhlin's definitions of the symbol rely on multidimensional Fourier transform theory.
A proof of Giraud's formulas was given later by Mikhlin ([6], §1.4, p. 9), and only later, in 1956 ([2], p. 54 and [8], §1.1.7 p. 11) he also proved that the symbol is the partial multidimensional Fourier transform respect to the integration variable of the singular integral, connecting his theory to the one Alberto Càlderon and Antoni Zygmund were developing in the same years.
References.
[1] Fichera, Gaetano, "Francesco Giacomo Tricomi", Atti della Accademia Nazionale dei Lincei. Serie VIII. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 66, pp. 469-483 (1979), MR0606447, Zbl 0463.01022.
[2] Fichera, Gaetano, "Solomon G. Mikhlin (1908-1990)", Atti della Accademia Nazionale dei Lincei, Rendiconti Lincei, Matematica e Applicazioni, Serie XI, Supplemento 5, pp. 49-61, 1 plate (1994), Zbl 0852.01034.
[3] Giraud, Georges, "Équations à intégrales principales; étude suivie d’une application", Annales Scientifiques de l'École Normale Supérieure. (3) 51, pp. 251-372 (1934), MR1509344, Zbl 0011.21604.
[4] Giraud, Georges, "Sur une classe générale d’équations à intégrales principales", Comptes rendus hebdomadaires des séances de l'Académie des sciences, Paris 202, pp. 2124-2127 (1936), JFM 62.0498.01, Zbl 0014.30903.
[5] Maz’ya, Vladimir, Differential equations of my young years. Translated from the Russian by Arkady Alexeev, Cham: Birkhäuser/Springer (ISBN 978-3-319-01808-9/hbk; 978-3-319-01809-6/ebook), pp. xiii+191 (2014), MR3288312, Zbl 1303.01002.
[6] Mikhlin, Solomon G., "Equations intégrales singulieres à deux variables independantes", Recueil Mathématique (Matematicheskii Sbornik) N.S. (in Russian), 1(43) (4), pp. 535-552 (1936), JFM 62.0495.02, Zbl 0016.02902.
[7] Mikhlin, Solomon G., Complément à l’article “Equations intégrales singulières à deux variables indépendantes”, Recueil Mathématique (Matematicheskii Sbornik) N.S. (in Russian), 1(43) (6), pp. 963-964 (1936), JFM 62.1251.02, Zbl 62.1251.02."
[8] Mikhlin, Solomon G., Multidimensional singular integrals and integral equations. Translated from the Russian by W. J. A. Whyte. Translation edited by I. N. Sneddon. (International Series of Monographs in Pure and Applied Mathematics. Vol. 83), Oxford-London-Edinburgh-New York-Paris-Frankfurt: Pergamon Press, pp. XII+255 (1965). MR0185399, ZBL0129.07701.
[9] Miranda, Carlo, Equazioni alle derivate parziali di tipo ellittico, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, 2. Heft., Berlin- Göttingen-Heidelberg: Springer-Verlag. VIII, 222 S. (1955), MR0087853, ZBL0065.08503.
[10] Miranda, Carlo, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Berlin-Heidelberg-New York: Springer-Verlag, pp. XII+370 (1970), MR0284700, Zbl 0198.14101.
[11] Tricomi, Francesco, "Formula d’inversione dell’ordine di due integrazioni doppie “con asterisco”"., Atti della Accademia Reale dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali, (6) 3, pp. 535-539 (1926), JFM 52.0235.05.
[12] Tricomi, Francesco, Equazioni integrali contenenti il valore principale di un integrale doppio, Mathematische Zeitschrift 27, pp. 87-133 (1927), JFM 53.0359.02.
Here is a practical problem in signal processing which is solved with pseudodifferential operators.
Let us assume that we want to take the derivative of some data represented by a function $u(t)$ where $t$ is the time variable. We could take the data in time into the frequency domain $U(\omega)$, then multiply by $-\mathrm{i} \omega$ and take the data back to the time domain. This can be represented as
\begin{eqnarray*} \frac{d u}{dt} = \int d \omega \; (-\mathrm{i} \omega ) U(\omega) \; \mathrm{e}^{\mathrm{i} \omega t} \end{eqnarray*} The left hand side is a differential operator. The right hand side is an example of a pseudodifferential operator with symbol $\sigma(\omega)= \mathrm{i} \omega$.
What if we want to compute the half derivative. That is, what would be the meaning of $d^{1/2} u/dx$? If in the frequency domain, a multiplication by $(-\mathrm{i} \omega)$ is a full derivative then a multiplication with $\sqrt{-\mathrm{i} \omega}$ in the frequency domain would be a half derivative. That is,
\begin{eqnarray} \frac{d^{1/2} u(t)}{dt} = \int d \omega \; (\sqrt{-\mathrm{i} \omega }) U(\omega) \; \mathrm{e}^{\mathrm{i} \omega t} \label{rho} \end{eqnarray} While the left handside does not show an obvious formula or computer implementation the right hand side shows a way to estimate the half derivative. Of course any fractional derivative can be computed in this way. In signal processing we know that the full derivative performs a 90 degree phase shift on each frequency component on the data. Sometimes, for problems of wave propagation in 2D, we need to perform a 45 degrees phase shift on each frequency component (the so called ``rho filter'' ) and this is achieved with the half derivative filter. The integral representation above is another example of a pseudodifferential operator where the symbol $\sqrt{-\mathrm{i} \omega}$ is not a polynomial in $\omega$.
