# Pseudo-differential operator

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Type of differential operator

In [mathematical analysis](/source/Mathematical_analysis) a **pseudo-differential operator** is an extension of the concept of [differential operator](/source/Differential_operator). Pseudo-differential operators are used extensively in the theory of [partial differential equations](/source/Partial_differential_equations) and [quantum field theory](/source/Quantum_field_theory), e.g. in mathematical models that include ultrametric [pseudo-differential equations](https://en.wikipedia.org/w/index.php?title=Pseudo-differential_equations&action=edit&redlink=1) in a [non-Archimedean](/source/Archimedean_property) space.

## History

The study of pseudo-differential operators began in the mid 1960s with the work of [Kohn](/source/Joseph_J._Kohn), [Nirenberg](/source/Louis_Nirenberg), [Hörmander](/source/Lars_H%C3%B6rmander), Unterberger and Bokobza.[1]

They played an influential role in the second proof of the [Atiyah–Singer index theorem](/source/Atiyah%E2%80%93Singer_index_theorem) via [K-theory](/source/Topological_K-theory). Atiyah and Singer thanked Hörmander for assistance with understanding the theory of pseudo-differential operators.[2]

## Motivation

### Linear differential operators with constant coefficients

Consider a linear [differential operator](/source/Differential_operator) with constant coefficients,

- P ( D ) := ∑ α a α D α {\displaystyle P(D):=\sum _{\alpha }a_{\alpha }\,D^{\alpha }}

which acts on smooth functions u {\displaystyle u} with compact support in **R***n*. This operator can be written as a composition of a [Fourier transform](/source/Fourier_transform), a simple *multiplication* by the polynomial function (called the **[symbol](/source/Fourier_multiplier)**)

- P ( ξ ) = ∑ α a α ξ α , {\displaystyle P(\xi )=\sum _{\alpha }a_{\alpha }\,\xi ^{\alpha },}

and an inverse Fourier transform, in the form:

P ( D ) u ( x ) = 1 ( 2 π ) n ∫ R n ∫ R n e i ( x − y ) ξ P ( ξ ) u ( y ) d y d ξ {\displaystyle \quad P(D)u(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}e^{i(x-y)\xi }P(\xi )u(y)\,dy\,d\xi } 1

Here, α = ( α 1 , … , α n ) {\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})} is a [multi-index](/source/Multi-index), a α {\displaystyle a_{\alpha }} are complex numbers, and

- D α = ( − i ∂ 1 ) α 1 ⋯ ( − i ∂ n ) α n {\displaystyle D^{\alpha }=(-i\partial _{1})^{\alpha _{1}}\cdots (-i\partial _{n})^{\alpha _{n}}}

is an iterated partial derivative, where ∂*j* means differentiation with respect to the *j*-th variable. We introduce the constants − i {\displaystyle -i} to facilitate the calculation of Fourier transforms.

**Derivation of formula (**[1](#math_1)**)**

The Fourier transform of a smooth function *u*, [compactly supported](/source/Compact_support) in **R***n*, is

- u ^ ( ξ ) := ∫ e − i y ξ u ( y ) d y {\displaystyle {\hat {u}}(\xi ):=\int e^{-iy\xi }u(y)\,dy}

and [Fourier's inversion formula](/source/Fourier's_inversion_formula) gives

- u ( x ) = 1 ( 2 π ) n ∫ e i x ξ u ^ ( ξ ) d ξ = 1 ( 2 π ) n ∬ e i ( x − y ) ξ u ( y ) d y d ξ {\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int e^{ix\xi }{\hat {u}}(\xi )d\xi ={\frac {1}{(2\pi )^{n}}}\iint e^{i(x-y)\xi }u(y)\,dy\,d\xi }

By applying *P*(*D*) to this representation of *u* and using

- P ( D x ) e i ( x − y ) ξ = e i ( x − y ) ξ P ( ξ ) {\displaystyle P(D_{x})\,e^{i(x-y)\xi }=e^{i(x-y)\xi }\,P(\xi )}

one obtains formula (**[1](#math_1)**).

### Representation of solutions to partial differential equations

To solve the partial differential equation

- P ( D ) u = f {\displaystyle P(D)\,u=f}

we (formally) apply the Fourier transform on both sides and obtain the *algebraic* equation

- P ( ξ ) u ^ ( ξ ) = f ^ ( ξ ) . {\displaystyle P(\xi )\,{\hat {u}}(\xi )={\hat {f}}(\xi ).}

If the symbol *P*(ξ) is never zero when ξ ∈ **R***n*, then it is possible to divide by *P*(ξ):

- u ^ ( ξ ) = 1 P ( ξ ) f ^ ( ξ ) {\displaystyle {\hat {u}}(\xi )={\frac {1}{P(\xi )}}{\hat {f}}(\xi )}

By Fourier's inversion formula, a solution is

- u ( x ) = 1 ( 2 π ) n ∫ e i x ξ 1 P ( ξ ) f ^ ( ξ ) d ξ . {\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int e^{ix\xi }{\frac {1}{P(\xi )}}{\hat {f}}(\xi )\,d\xi .}

Here it is assumed that:

1. *P*(*D*) is a linear differential operator with *constant* coefficients,

1. its symbol *P*(ξ) is never zero,

1. both *u* and ƒ have a well defined Fourier transform.

The last assumption can be weakened by using the theory of [distributions](/source/Distribution_(mathematics)). The first two assumptions can be weakened as follows.

In the last formula, write out the Fourier transform of ƒ to obtain

- u ( x ) = 1 ( 2 π ) n ∬ e i ( x − y ) ξ 1 P ( ξ ) f ( y ) d y d ξ . {\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\iint e^{i(x-y)\xi }{\frac {1}{P(\xi )}}f(y)\,dy\,d\xi .}

This is similar to formula (**[1](#math_1)**), except that 1/*P*(ξ) is not a polynomial function, but a function of a more general kind.

## Definition of pseudo-differential operators

Here we view pseudo-differential operators as a generalization of differential operators. We extend formula (1) as follows. A **pseudo-differential operator** *P*(*x*,*D*) on **R***n* is an operator whose value on the function *u(x)* is the function of *x*:

P ( x , D ) u ( x ) = 1 ( 2 π ) n ∫ R n e i x ⋅ ξ P ( x , ξ ) u ^ ( ξ ) d ξ {\displaystyle \quad P(x,D)u(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }P(x,\xi ){\hat {u}}(\xi )\,d\xi } 2

where u ^ ( ξ ) {\displaystyle {\hat {u}}(\xi )} is the [Fourier transform](/source/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

- | ∂ ξ α ∂ x β P ( x , ξ ) | ≤ C α , β ( 1 + | ξ | ) m − | α | {\displaystyle |\partial _{\xi }^{\alpha }\partial _{x}^{\beta }P(x,\xi )|\leq C_{\alpha ,\beta }\,(1+|\xi |)^{m-|\alpha |}}

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 S 1 , 0 m {\displaystyle \scriptstyle {S_{1,0}^{m}}} of [Hörmander](/source/H%C3%B6rmander). The corresponding operator *P*(*x*,*D*) is called a **pseudo-differential operator of order m** and belongs to the class Ψ 1 , 0 m . {\displaystyle \Psi _{1,0}^{m}.}

## Properties

Linear differential operators of order m with smooth bounded coefficients are pseudo-differential operators of order *m*. The composition *PQ* of two pseudo-differential operators *P*, *Q* is again a pseudo-differential operator and the symbol of *PQ* can be calculated by using the symbols of *P* and *Q*. The adjoint and transpose of a pseudo-differential operator is a pseudo-differential operator.

If a differential operator of order *m* is [(uniformly) elliptic](/source/Elliptic_differential_operator) (of order *m*) and invertible, then its inverse is a pseudo-differential 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 pseudo-differential 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. Pseudo-differential operators are *pseudo-local*, which means informally that when applied to a [distribution](/source/Schwartz_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/d*x* in the form

- p ( x , D ) {\displaystyle p(x,D)\,}

for a [polynomial](/source/Polynomial) *p* in *D* (which is called the *symbol*), a pseudo-differential operator has a symbol in a more general class of functions. Often one can reduce a problem in analysis of pseudo-differential operators to a sequence of algebraic problems involving their symbols, and this is the essence of [microlocal analysis](/source/Microlocal_analysis).

## Kernel of pseudo-differential operator

Pseudo-differential operators can be represented by [kernels](/source/Integral_transform). 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](/source/Singular_integral).

## See also

- [Differential algebra](/source/Differential_algebra) for a definition of pseudo-differential operators in the context of differential algebras and differential rings.

- [Fourier transform](/source/Fourier_transform)

- [Fourier integral operator](/source/Fourier_integral_operator)

- [Oscillatory integral operator](/source/Oscillatory_integral_operator)

- [Sato's fundamental theorem](https://en.wikipedia.org/w/index.php?title=Sato%27s_fundamental_theorem&action=edit&redlink=1)

- [Operational calculus](/source/Operational_calculus)

- [Microdifferential operator](/source/Microdifferential_operator)

## Footnotes

1. **[^](#cite_ref-1)** [Stein 1993](#CITEREFStein1993), Chapter 6

1. **[^](#cite_ref-2)** [Atiyah & Singer 1968](#CITEREFAtiyahSinger1968), p. 486

## References

- [Stein, Elias](/source/Elias_Stein) (1993), *Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals*, Princeton University Press.

- [Atiyah, Michael F.](/source/Michael_Atiyah); [Singer, Isadore M.](/source/Isadore_Singer) (1968), "The Index of Elliptic Operators I", *Annals of Mathematics*, **87** (3): 484–530, [doi](/source/Doi_(identifier)):[10.2307/1970715](https://doi.org/10.2307%2F1970715), [JSTOR](/source/JSTOR_(identifier)) [1970715](https://www.jstor.org/stable/1970715)

## Further reading

- Nicolas Lerner, *Metrics on the phase space and non-selfadjoint pseudo-differential operators*. Pseudo-Differential Operators. Theory and Applications, 3. Birkhäuser Verlag, Basel, 2010.

- [Michael E. Taylor](/source/Michael_E._Taylor), Pseudodifferential Operators, Princeton Univ. Press 1981. [ISBN](/source/ISBN_(identifier)) [0-691-08282-0](https://en.wikipedia.org/wiki/Special:BookSources/0-691-08282-0)

- M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag 2001. [ISBN](/source/ISBN_(identifier)) [3-540-41195-X](https://en.wikipedia.org/wiki/Special:BookSources/3-540-41195-X)

- [Francois Treves](/source/Francois_Treves), Introduction to Pseudo Differential and Fourier Integral Operators, (University Series in Mathematics), Plenum Publ. Co. 1981. [ISBN](/source/ISBN_(identifier)) [0-306-40404-4](https://en.wikipedia.org/wiki/Special:BookSources/0-306-40404-4)

- F. G. Friedlander and M. Joshi, Introduction to the Theory of Distributions, Cambridge University Press 1999. [ISBN](/source/ISBN_(identifier)) [0-521-64971-4](https://en.wikipedia.org/wiki/Special:BookSources/0-521-64971-4)

- [Hörmander, Lars](/source/Lars_H%C3%B6rmander) (1987). *The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators*. Springer. [ISBN](/source/ISBN_(identifier)) [3-540-49937-7](https://en.wikipedia.org/wiki/Special:BookSources/3-540-49937-7).

- André Unterberger, *Pseudo-differential operators and applications: an introduction*. Lecture Notes Series, 46. Aarhus Universitet, Matematisk Institut, Aarhus, 1976.

## External links

- [Lectures on Pseudo-differential Operators](https://arxiv.org/abs/math.AP/9906155) by [Mark S. Joshi](/source/Mark_S._Joshi) on arxiv.org.

- ["Pseudo-differential operator"](https://www.encyclopediaofmath.org/index.php?title=Pseudo-differential_operator), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society), 2001 [1994]

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