In mathematical analysis a pseudodifferential operator is an extension of the concept of differential operator. Pseudodifferential operators are used extensively in the theory of partial differential equations and quantum field theory.
History
The study of pseudodifferential operators began in the mid 1960s with the work of Kohn, Nirenberg, Hörmander, Unterberger and Bokobza.^{[1]}
They played an influential role in the second proof of the Atiyah–Singer index theorem via Ktheory. Atiyah and Singer thanked Hörmander for assistance with understanding the theory of Pseudodifferential operators.^{[2]}
Motivation
Linear differential operators with constant coefficients
Consider a linear differential operator with constant coefficients,
which acts on smooth functions with compact support in R^{n}. This operator can be written as a composition of a Fourier transform, a simple multiplication by the polynomial function (called the symbol)
and an inverse Fourier transform, in the form:

(1)
Here, is a multiindex, are complex numbers, and
is an iterated partial derivative, where ∂_{j} means differentiation with respect to the jth variable. We introduce the constants to facilitate the calculation of Fourier transforms.
 Derivation of formula (1)
The Fourier transform of a smooth function u, compactly supported in R^{n}, is
and Fourier's inversion formula gives
By applying P(D) to this representation of u and using
one obtains formula (1).
Representation of solutions to partial differential equations
To solve the partial differential equation
we (formally) apply the Fourier transform on both sides and obtain the algebraic equation
If the symbol P(ξ) is never zero when ξ ∈ R^{n}, then it is possible to divide by P(ξ):
By Fourier's inversion formula, a solution is
Here it is assumed that:
 P(D) is a linear differential operator with constant coefficients,
 its symbol P(ξ) is never zero,
 both u and ƒ have a well defined Fourier transform.
The last assumption can be weakened by using the theory of distributions. The first two assumptions can be weakened as follows.
In the last formula, write out the Fourier transform of ƒ to obtain
This is similar to formula (1), except that 1/P(ξ) is not a polynomial function, but a function of a more general kind.
Definition of pseudodifferential operators
Here we view pseudodifferential operators as a generalization of differential operators. We extend formula (1) as follows. A pseudodifferential operator P(x,D) on R^{n} is an operator whose value on the function u(x) is the function of x:

(2)
where is the Fourier transform of u and the symbol P(x,ξ) in the integrand belongs to a certain symbol class. For instance, if P(x,ξ) is an infinitely differentiable function on R^{n} × R^{n} with the property
for all x,ξ ∈R^{n}, all multiindices α,β, some constants C_{α, β} and some real number m, then P belongs to the symbol class of Hörmander. The corresponding operator P(x,D) is called a pseudodifferential operator of order m and belongs to the class
Properties
Linear differential operators of order m with smooth bounded coefficients are pseudodifferential operators of order m. The composition PQ of two pseudodifferential operators P, Q is again a pseudodifferential operator and the symbol of PQ can be calculated by using the symbols of P and Q. The adjoint and transpose of a pseudodifferential operator is a pseudodifferential operator.
If a differential operator of order m is (uniformly) elliptic (of order m) and invertible, then its inverse is a pseudodifferential operator of order −m, and its symbol can be calculated. This means that one can solve linear elliptic differential equations more or less explicitly by using the theory of pseudodifferential operators.
Differential operators are local in the sense that one only needs the value of a function in a neighbourhood of a point to determine the effect of the operator. Pseudodifferential operators are pseudolocal, which means informally that when applied to a distribution they do not create a singularity at points where the distribution was already smooth.
Just as a differential operator can be expressed in terms of D = −id/dx in the form
for a polynomial p in D (which is called the symbol), a pseudodifferential operator has a symbol in a more general class of functions. Often one can reduce a problem in analysis of pseudodifferential operators to a sequence of algebraic problems involving their symbols, and this is the essence of microlocal analysis.
Kernel of pseudodifferential operator
Pseudodifferential operators can be represented by kernels. The singularity of the kernel on the diagonal depends on the degree of the corresponding operator. In fact, if the symbol satisfies the above differential inequalities with m ≤ 0, it can be shown that the kernel is a singular integral kernel.
See also
 Differential algebra for a definition of pseudodifferential operators in the context of differential algebras and differential rings.
 Fourier transform
 Fourier integral operator
 Oscillatory integral operator
 Sato's fundamental theorem
Footnotes
 ^ Stein 1993, Chapter 6
 ^ Atiyah & Singer 1968, p. 486
References
 Stein, Elias (1993), Harmonic Analysis: RealVariable Methods, Orthogonality and Oscillatory Integrals, Princeton University PressCS1 maint: ref=harv (link).
 Atiyah, Michael F.; Singer, Isadore M. (1968), "The Index of Elliptic Operators I", Annals of Mathematics, 87 (3): 484–530, doi:10.2307/1970715, JSTOR 1970715CS1 maint: ref=harv (link)
Further reading
 Michael E. Taylor, Pseudodifferential Operators, Princeton Univ. Press 1981. ISBN 0691082820
 M. A. Shubin, Pseudodifferential Operators and Spectral Theory, SpringerVerlag 2001. ISBN 354041195X
 Francois Treves, Introduction to Pseudo Differential and Fourier Integral Operators, (University Series in Mathematics), Plenum Publ. Co. 1981. ISBN 0306404044
 F. G. Friedlander and M. Joshi, Introduction to the Theory of Distributions, Cambridge University Press 1999. ISBN 0521649714
 Hörmander, Lars (1987). The Analysis of Linear Partial Differential Operators III: PseudoDifferential Operators. Springer. ISBN 3540499377.
External links
 Lectures on Pseudodifferential Operators by Mark S. Joshi on arxiv.org.
 "Pseudodifferential operator", Encyclopedia of Mathematics, EMS Press, 2001 [1994]