


  • 14
    $\begingroup$ 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! $\endgroup$ – Mariano Suárez-Álvarez Mar 21 '11 at 18:30
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    $\begingroup$ 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.) $\endgroup$ – shuhalo Mar 21 '11 at 18:44
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    $\begingroup$ 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) $\endgroup$ – Camilo Sarmiento Apr 9 '13 at 21:38

10 Answers 10














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    $\begingroup$ I would add between (2) and (3) the introduction of the symbolic calculus of pseudos by Kohn and Nirenberg (1964 - classical, polyhomogeneous symbols of type $(1,0)$) and Hörmander (1965 - standard symbols of type $(\rho,\delta)$), as Deane Yang remarked in his answer (the year difference is just a formal matter- as timur commented there, both works appeared in the same volume of CPAM). Another addition would be the Hörmander-Weyl symbolic calculus (1979) after a much earlier idea of Weyl (1928) in the context of quantum mechanics, generalizing the symbolic calculus of Beals and Fefferman. $\endgroup$ – Pedro Lauridsen Ribeiro Jun 18 '16 at 2:10


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


3) the inverse of an elliptic operator;

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


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:



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    $\begingroup$ The same volume of CPAM 1965 also contains a paper of Hörmander that defines pseudodifferential operators (in a manifestly coordinate invariant way). $\endgroup$ – timur Jun 1 '11 at 3:10
  • $\begingroup$ timur, thanks for the additional info. $\endgroup$ – Deane Yang May 22 '12 at 8:50
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    $\begingroup$ The paper of Lax was important in a broader context - there we find the first (local) prototype of Fourier integral operators, since parametrices of hyperbolic PDE are actually not pseudodifferential. Nonetheless, the idea of an asymptotic expansion to build a parametrix was indeed the precursor of the symbolic calculus of Kohn and Nirenberg (one should perhaps add independent work by Bokobza and Unterberger). $\endgroup$ – Pedro Lauridsen Ribeiro Jun 18 '16 at 2:16
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    $\begingroup$ I believe that the earliest instance of a symbolic calculus involving variable-coefficient operators was due to Hermann Weyl in his book "Group Theory and Quantum Mechanics", whose earliest (German) edition dates from 1928. The idea was more than 40 years later implemented rigorously by Hörmander (1979) using a far more general class of symbols, after earlier work by Grossman, Loupias and Stein (1969), Berezin and Shubin (1970) and Voros (1976). $\endgroup$ – Pedro Lauridsen Ribeiro Jun 18 '16 at 3:16







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.






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

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

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 $\omega$.


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.


[1] Fichera, Gaetano, "Francesco Giacomo Tricomi", Atti della Accademia Nazionale dei Lincei. Serie VIII. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 66, 469-483 (1979). MR0606447, ZBL0463.01022.

[2] Fichera, Gaetano, "Solomon G. Mikhlin (1908-1990)", Atti della Accademia Nazionale dei Lincei, Rendiconti Lincei, Matematica e Applicazioni, Serie XI, Supplemento 5, 49-61, 1 plate (1994). ZBL0852.01034.

[3] Giraud, Georges, "Équations à intégrales principales; étude suivie d’une application", Annales Scientifiques de l'École Normale Supérieure. (3) 51, 251-372 (1934). MR1509344, ZBL0011.21604.

[4] Giraud, Georges, "Sur une classe générale d’équations à intégrales principales", Comptes rendus hebdomadaires des séances de l'Académie des sciences, Paris 202, 2124-2127 (1936). JFM62.0498.01, ZBL0014.30903.

[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). xiii, 191 p. (2014). MR3288312 ZBL1303.01002.

[6] Mikhlin, Solomon G., "Equations intégrales singulieres à deux variables independantes", Recueil Mathématique (Matematicheskii Sbornik) N.S. (in Russian), 1(43) (4), 535-552 (1936). JFM62.0495.02, ZBL0016.02902.

[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), 963-964 (1936). JFM62.1251.02, ZBL62.1251.02."

[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. XII, 255 p. (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 XII, 370 p. (1970), MR0284700, ZBL0198.14101.

[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, 535-539 (1926). JFM 52.0235.05.

[12] Tricomi, Francesco, Equazioni integrali contenenti il valore principale di un integrale doppio, Mathematische Zeitschrift 27, 87-133 (1927). JFM 53.0359.02.


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