Editing Pseudo-differential operator

{{Short description|Type of differential operator}}
In [[mathematical analysis]] a '''pseudo-differential operator'''  is an extension of the concept of [[differential operator]]. Pseudo-differential operators are used extensively in the theory of [[partial differential equations]] and [[quantum field theory]], e.g. in mathematical models that include ultrametric [[pseudo-differential equations]] in a [[Archimedean property|non-Archimedean]] space.

==History==
The study of pseudo-differential operators began in the mid 1960s with the work of [[Joseph J. Kohn|Kohn]], [[Louis Nirenberg|Nirenberg]], [[Lars Hörmander|Hörmander]], Unterberger and Bokobza.<ref>{{harvnb|Stein|1993|loc=Chapter 6}}</ref>

They played an influential role in the second proof of the [[Atiyah–Singer index theorem]] via [[Topological K-theory|K-theory]].  Atiyah and Singer thanked Hörmander for assistance with understanding the theory of pseudo-differential operators.<ref>{{harvnb|Atiyah|Singer|1968|page=486}}</ref>

==Motivation==

===Linear differential operators with constant coefficients===
Consider a linear [[differential operator]] with constant coefficients,

:<math> P(D) := \sum_\alpha a_\alpha \, D^\alpha </math>

which acts on smooth functions <math>u</math> with compact support in '''R'''<sup>''n''</sup>.
This operator can be written as a composition of a [[Fourier transform]], a simple ''multiplication'' by the
polynomial function (called the '''[[Fourier multiplier|symbol]]''')

:<math>  P(\xi) = \sum_\alpha a_\alpha \, \xi^\alpha, </math>

and an inverse Fourier transform, in the form:

{{NumBlk|:|<math> \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 </math>|{{EquationRef|1}}}}

Here, <math>\alpha = (\alpha_1,\ldots,\alpha_n)</math> is a [[multi-index]], <math>a_\alpha</math> are complex numbers, and

:<math>D^\alpha=(-i \partial_1)^{\alpha_1} \cdots (-i \partial_n)^{\alpha_n}</math>

is an iterated partial derivative, where ∂<sub>''j''</sub> means differentiation with respect to the ''j''-th variable.  We introduce the constants <math>-i</math> to facilitate the calculation of Fourier transforms.

;Derivation of formula ({{EquationNote|1}})
The Fourier transform of a smooth function ''u'', [[compact support|compactly supported]] in '''R'''<sup>''n''</sup>, is

:<math>\hat u (\xi) := \int e^{- i y \xi} u(y) \, dy</math>

and [[Fourier's inversion formula]] gives

:<math>u (x) = \frac{1}{(2 \pi)^n} \int e^{i x \xi} \hat u (\xi) d\xi = 
\frac{1}{(2 \pi)^n} \iint e^{i (x - y) \xi} u (y) \, dy \, d\xi </math>

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

:<math>P(D_x) \, e^{i (x - y) \xi} = e^{i (x - y) \xi} \, P(\xi) </math>

one obtains formula ({{EquationNote|1}}).

===Representation of solutions to partial differential equations===

To solve the partial differential equation

:<math> P(D) \, u = f </math>

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

:<math> P(\xi) \, \hat u (\xi) = \hat f(\xi). </math>

If the symbol ''P''(&xi;) is never zero when &xi;&nbsp;&isin;&nbsp;'''R'''<sup>''n''</sup>, then it is possible to divide by ''P''(&xi;):

:<math> \hat u(\xi) = \frac{1}{P(\xi)} \hat f(\xi) </math>

By Fourier's inversion formula, a solution is

:<math>  u (x) = \frac{1}{(2 \pi)^n} \int e^{i x \xi} \frac{1}{P(\xi)} \hat f (\xi) \, d\xi.</math>

Here it is assumed that:
# ''P''(''D'') is a linear differential operator with ''constant'' coefficients,
# its symbol ''P''(&xi;) is never zero,
# both ''u'' and &fnof; have a well defined Fourier transform.
The last assumption can be weakened by using the theory of [[distribution (mathematics)|distribution]]s.
The first two assumptions can be weakened as follows.

In the last formula, write out the Fourier transform of &fnof; to obtain

:<math>  u (x) = \frac{1}{(2 \pi)^n} \iint e^{i (x-y) \xi} \frac{1}{P(\xi)} f (y) \, dy \, d\xi.</math>

This is similar to formula ({{EquationNote|1}}), except that 1/''P''(&xi;) 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'''<sup>''n''</sup> is an operator whose value on the function ''u(x)'' is the function of ''x'':

{{NumBlk|:|<math>\quad P(x,D) u (x) = 
\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} e^{i x\cdot \xi} P(x,\xi) \hat{u}(\xi) \, d\xi </math>|{{EquationRef|2}}}}

where <math>\hat{u}(\xi)</math> is the [[Fourier transform]] of ''u'' and the symbol ''P''(''x'',&xi;) in the integrand belongs to a certain ''symbol class''.
For instance, if ''P''(''x'',&xi;) is an infinitely differentiable function on '''R'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''R'''<sup>''n''</sup> with the property

:<math> |\partial_\xi^\alpha \partial_x^\beta P(x,\xi)| \leq C_{\alpha,\beta} \, (1 + |\xi|)^{m - |\alpha|} </math>

for all ''x'',&xi;&nbsp;&isin;'''R'''<sup>''n''</sup>, all multiindices &alpha;,&beta;, some constants ''C''<sub>&alpha;, &beta;</sub> and some real number ''m'', then ''P'' belongs to the symbol class <math>\scriptstyle{S^m_{1,0}}</math> of [[Hörmander]]. The corresponding operator ''P''(''x'',''D'') is called a '''pseudo-differential operator of order m''' and belongs to the class
<math>\Psi^m_{1,0}.</math>

==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'',&nbsp;''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 [[elliptic differential operator|(uniformly) elliptic]] (of order ''m'')
and invertible, then its inverse is a pseudo-differential operator of order &minus;''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 [[Schwartz distribution|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''&nbsp;=&nbsp;&minus;id/d''x'' in the form

:<math>p(x, D)\,</math>

for a [[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]].

==Kernel of pseudo-differential operator==

Pseudo-differential operators can be represented by [[Integral transform|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|singular integral kernel]]. <!--The kernels are used for characterization of boundary data for inverse boundary problems.-->

==See also==
* [[Differential algebra]] for a definition of pseudo-differential operators in the context of differential algebras and differential rings.
* [[Fourier transform]]
* [[Fourier integral operator]]
* [[Oscillatory integral operator]]
* [[Sato's fundamental theorem]]
* [[Operational calculus]]

==Footnotes==
{{Reflist}}

==References==
* {{citation|first=Elias|last=Stein|authorlink=Elias Stein|title=Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals|publisher=Princeton University Press|year=1993}}.
* {{citation|last1= Atiyah|first1= Michael F. |author1-link=Michael Atiyah|last2=Singer|first2= Isadore M. |author2-link=Isadore Singer|title=The Index of Elliptic Operators I|journal= Annals of Mathematics |volume=87|pages= 484–530|year= 1968|doi= 10.2307/1970715|issue= 3|jstor=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]], Pseudodifferential Operators, Princeton Univ. Press 1981. {{ISBN|0-691-08282-0}}
* M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag 2001. {{ISBN|3-540-41195-X}}
* [[Francois Treves]], Introduction to Pseudo Differential and Fourier Integral Operators, (University Series in Mathematics), Plenum Publ. Co. 1981. {{ISBN|0-306-40404-4}}
*  F. G. Friedlander and M. Joshi, Introduction to the Theory of Distributions, Cambridge University Press 1999. {{ISBN|0-521-64971-4}}
* {{cite book
|first=Lars
|last=Hörmander
|authorlink= Lars Hörmander
|title=The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators
|year=1987
|publisher=Springer
|isbn=3-540-49937-7}}
<!--
* Ingerman D.V. and Morrow J.A., [http://www.math.washington.edu/~morrow/papers/imrev.pdf "On a characterization of the kernel of the Dirichlet-to-Neumann map for a planar region"], ''SIAM J. Math. Anal.'' 1998, vol.&nbsp;29, no.&nbsp;1, pp.&nbsp;106–115 (electronic).
-->
* André Unterberger, ''Pseudo-differential operators and applications: an introduction''. Lecture Notes Series, 46. Aarhus Universitet, Matematisk Institut, Aarhus, 1976. 
==External links==
* [https://arxiv.org/abs/math.AP/9906155 Lectures on Pseudo-differential Operators] by [[Mark S. Joshi]] on arxiv.org.
* {{springer|title=Pseudo-differential operator|id=p/p075660}}

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[[Category:Differential operators]]
[[Category:Microlocal analysis]]
[[Category:Functional analysis]]
[[Category:Harmonic analysis]]
[[Category:Generalized functions]]
[[Category:Partial differential equations]]