**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, 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.