Pseudo differential operator

A pseudo differential operator is an extension of the concept of the differential operator . They are an important part of the theory of partial differential equations . The basics of the theory come from Lars Hörmander . They were introduced in 1965 by Joseph Kohn and Louis Nirenberg . Your studies form an important part of micro-local analysis .

motivation

Linear differential operators with constant coefficients

Consider the linear differential operator with constant coefficients

${\displaystyle p(D):=\sum _{\alpha }a_{\alpha }\,D^{\alpha }}$

who operates in the space of smooth functions with compact vehicle in . It can be used as a composition of a Fourier transform , a simple multiplication with the polynomial (the so-called symbol ) ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle p(\xi )=\sum _{\alpha }a_{\alpha }\,\xi ^{\alpha }}$

and the inverse Fourier transform:

${\displaystyle (1)\quad p(D)u(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }p(\xi )u(y)\,\mathrm {d} y\,\mathrm {d} \xi }$

to be written. It is a multi-index , a differential operator, represents the derivative with respect to th component and are complex numbers. ${\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})\in \mathbb {N} _{0}^{n}}$ ${\displaystyle D^{\alpha }=(-i\partial _{1})^{\alpha _{1}}\dots (-i\partial _{n})^{\alpha _{n}}}$ ${\displaystyle \partial _{j}}$ ${\displaystyle j}$ ${\displaystyle a_{\alpha }\,}$

Analog is a pseudo differential operator with a symbol on an operator of the form ${\displaystyle P}$ ${\displaystyle p(x,\xi )}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle (2)\quad Pu(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }p(x,\xi )u(y)\,\mathrm {d} y\,\mathrm {d} \xi }$,

with a more general function in the integrand, as explained below. ${\displaystyle p}$

Derivation of formula (1)

The Fourier transform of a smooth function , with compact support in , is ${\displaystyle u}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle {\hat {u}}(\xi ):=\int \mathrm {e} ^{-\mathrm {i} y\xi }u(y)\,\mathrm {d} y}$

and inverse Fourier transform results

${\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int \mathrm {e} ^{\mathrm {i} x\xi }{\hat {u}}(\xi )\,\mathrm {d} \xi ={\frac {1}{(2\pi )^{n}}}\int \int \mathrm {e} ^{\mathrm {i} (x-y)\xi }u(y)\,\mathrm {d} y\,\mathrm {d} \xi \,.}$

If one applies to this representation of and uses ${\displaystyle p(D)}$ ${\displaystyle u}$

${\displaystyle p(D)\,\mathrm {e} ^{\mathrm {i} (x-y)\xi }=\mathrm {e} ^{\mathrm {i} (x-y)\xi }\,p(\xi )}$

one obtains (1).

Representation of solutions to partial differential equations

To a partial differential equation

${\displaystyle p(D)\,u=f}$

to solve both sides are (formally) Fourier transformed, whereby algebraic equations result:

${\displaystyle p(\xi )\,{\hat {u}}(\xi )={\hat {f}}(\xi )}$.

If the symbol is always not equal to zero for , one can use : ${\displaystyle p(\xi )}$ ${\displaystyle \xi \in \mathbb {R} ^{n}}$ ${\displaystyle p(\xi )}$

to divide:

${\displaystyle {\hat {u}}(\xi )={\frac {1}{p(\xi )}}{\hat {f}}(\xi )}$

Using the inverse Fourier transform, the solution is:

${\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int \mathrm {e} ^{\mathrm {i} x\xi }{\frac {1}{p(\xi )}}{\hat {f}}(\xi )\,\mathrm {d} \xi }$.

The following is assumed:

1. ${\displaystyle p(D)}$ is a linear differential operator with constant coefficients,
2. its symbol is never zero for , ${\displaystyle p(\xi )}$ ${\displaystyle \xi \neq 0}$
3. both and have well-defined Fourier transforms. ${\displaystyle u}$ ${\displaystyle f}$

The latter assumption can be weakened with the theory of distributions . The first two assumptions can be weakened as follows:

Insert the Fourier transform of f in the last formula :

${\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int \int \mathrm {e} ^{\mathrm {i} (x-y)\xi }{\frac {1}{p(\xi )}}f(y)\,\mathrm {d} y\,\mathrm {d} \xi }$.

This is similar to formula (1), only that is not a polynomial, but a function of a more general kind. ${\displaystyle {\tfrac {1}{p(\xi )}}}$

Definition of the pseudo differential operator

The symbol class

→ Main article : Symbol class

Is an infinitely often differentiable function on , open, with ${\displaystyle a(x,\xi )}$ ${\displaystyle \Omega \times \mathbb {R} ^{n}}$ ${\displaystyle \Omega }$

${\displaystyle |\partial _{\xi }^{\alpha }\partial _{x}^{\beta }a(x,\xi )|\leq C_{\alpha ,\beta ,K}\,(1+|\xi |)^{m-|\alpha |}}$

for all , where is compact, for all , all multi-indices , a constant and real numbers m, then a belongs to the symbol class . ${\displaystyle x\in K}$ ${\displaystyle K\subset \Omega }$ ${\displaystyle \xi }$ ${\displaystyle \alpha ,\beta }$ ${\displaystyle C_{\alpha ,\beta ,K}}$ ${\displaystyle S^{m}(\Omega \times \mathbb {R} ^{n})}$

Pseudo differential operator

Again be a smooth function from the symbol class with . A pseudo differential operator of order m is usually a map ${\displaystyle a}$ ${\displaystyle S^{m}(X\times \mathbb {R} ^{n})}$ ${\displaystyle X\subset \mathbb {R} ^{n}}$

${\displaystyle {\mathcal {D}}(X)\to {\mathcal {E}}(X)\quad {\text{bzw.}}\quad {\mathcal {S}}(X)\to {\mathcal {S}}(X),}$

which through

${\displaystyle (Pu)(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }a(x,\xi )u(y)\,\mathrm {d} y\,\mathrm {d} \xi }$

is defined. The space is the space of the test functions , is the space of the smooth functions and is the Schwartz space . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {S}}}$

Actually worn pseudo differential operator

Let be a pseudo differential operator. The following is ${\displaystyle P}$

${\displaystyle K_{P}(x,y):=\int _{\mathbb {R} ^{n}}\mathrm {e} ^{\mathrm {i} (x-y)\xi }a(x,\xi )\,\mathrm {d} \xi }$

the integral kernel of the operator . The pseudo differential operator is actually called carried, if the projections are actually . ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle \pi _{1},\pi _{2}:\operatorname {supp} (K_{P})\to X}$

properties

• Linear differential operators of order m with smooth, bounded coefficients can be regarded as pseudo differential operators of order m.

• The integral core

${\displaystyle K(x,y):=\int _{\mathbb {R} ^{n}}^{OS}\mathrm {e} ^{\mathrm {i} \langle x-y,\xi \rangle }a(x,\xi )\,\mathrm {d} \xi }$

is a smooth black core except on the diagonal . ${\displaystyle \{(x,y)|x=y\}}$

• The transpose of a pseudo differential operator is also a pseudo differential operator.

• If a linear differential operator of order m is elliptic , its inverse is a pseudo differential operator of order -m. One can therefore solve linear, elliptical differential equations more or less explicitly with the help of the theory of pseudo differential operators.

• Differential operators are local . This means that one only needs to know the value of a function in the vicinity of a point in order to determine the effect of the operator. Pseudo differential operators are pseudo- local , which means that they do not increase the singular carrier of a distribution. So it applies

${\displaystyle \operatorname {sing\,supp} (Au)\subset \operatorname {sing\,supp} (u)}$.

• Since the Schwartz space lies close to the space of the square-integrable functions , it is possible to continue a pseudo differential operator by using continuity arguments . If it also holds, then is a bounded and continuous operator. ${\displaystyle L^{2}}$ ${\displaystyle L^{2}}$ ${\displaystyle A\in \Psi ^{0}(\mathbb {R} ^{n}\times \mathbb {R} ^{n})}$ ${\displaystyle A\colon L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})}$

Composition of pseudo differential operators

Pseudo differential operators with the Schwartz space as the domain of definition represent this in themselves. They are even based on an isomorphism . In addition, actually carried pseudo differential operators map the space in itself. It is therefore possible to consider the composition of two pseudo differential operators for such operators , which again results in a pseudo differential operator. ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$ ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$ ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$

Let and two symbols and let and be the corresponding pseudo differential operators, then again is a pseudo differential operator. The symbol of the operator is an element of space and it has the asymptotic expansion ${\displaystyle a\in S^{m_{1}}(X\times \mathbb {R} ^{n})}$ ${\displaystyle b\in S^{m_{2}}(X\times \mathbb {R} ^{n})}$ ${\displaystyle P_{a}}$ ${\displaystyle P_{b}}$ ${\displaystyle P_{a}\circ P_{b}}$ ${\displaystyle c}$ ${\displaystyle P_{a}\circ P_{b}}$ ${\displaystyle S^{m_{1}+m_{2}}(X\times \mathbb {R} ^{n})}$

${\displaystyle c\sim \sum _{\mu =0}^{\infty }{\frac {(-i)^{|\mu |}}{\mu !}}{\frac {\partial ^{\mu }a}{\partial \xi ^{\mu }}}(x,\xi ){\frac {\partial ^{\mu }b}{\partial x^{\mu }}}(x,\xi )\,,}$

What

${\displaystyle c-\sum _{\mu <N}{\frac {(-i)^{|\mu |}}{\mu !}}{\frac {\partial ^{\mu }a}{\partial \xi ^{\mu }}}(x,\xi ){\frac {\partial ^{\mu }b}{\partial x^{\mu }}}(x,\xi )\in S^{m_{1}+m_{2}-N}(X\times \mathbb {R} ^{n})}$

means.

Adjoint operator

For every pair of Schwartz functions let ${\displaystyle \phi ,\,\psi }$

${\displaystyle (\phi ,\psi )=\int _{X}\phi (x){\overline {\psi (x)}}\mathrm {d} x}$

be a bilinear form and be a pseudo differential operator with symbol . Then is formally adjoint operator with respect to another pseudo-differential operator and its symbol is an element of the space and it has the asymptotic expansion ${\displaystyle P\colon {\mathcal {S}}(X)\to {\mathcal {S}}(X)}$ ${\displaystyle a\in S^{m}(X\times \mathbb {R} ^{n})}$ ${\displaystyle P^{*}}$ ${\displaystyle (\cdot ,\cdot )}$ ${\displaystyle a^{*}}$ ${\displaystyle S^{m}(X\times \mathbb {R} ^{n})}$

${\displaystyle a^{*}\sim \sum _{\mu =0}^{\infty }{\frac {(-i)^{|\mu |}}{\mu !}}\,{\frac {\partial ^{\mu }}{\partial x^{\mu }}}{\frac {\partial ^{\mu }b}{\partial \xi ^{\mu }}}\,{\overline {a(x,\xi )}}\,.}$

Pseudo differential operators on distribution spaces

With the help of the formally adjoint operator it is possible to define pseudo differential operators on distribution spaces. For this purpose, instead of the bilinear form, one considers the dual pairing between the Schwartz space and its dual space . The dual pairing can be understood as a continuous continuation of . Therefore it is possible to define pseudo differential operators on the dual space of the Schwartz space, i.e. the space of the tempered distributions . ${\displaystyle (\cdot ,\cdot )}$ ${\displaystyle (\cdot ,\cdot )_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}}$ ${\displaystyle (\cdot ,\cdot )}$

Let be a pseudo differential operator and a tempered distribution. Then the continued operator for all is defined by ${\displaystyle P\colon {\mathcal {S}}(\mathbb {R} ^{n})\to {\mathcal {S}}(\mathbb {R} ^{n})}$ ${\displaystyle u\in {\mathcal {S}}'(\mathbb {R} ^{n})}$ ${\displaystyle {\tilde {P}}\colon {\mathcal {S}}'(\mathbb {R} ^{n})\to {\mathcal {S}}'(\mathbb {R} ^{n})}$ ${\displaystyle v\in {\mathcal {S}}(\mathbb {R} ^{n})}$

${\displaystyle ({\tilde {P}}u,v)_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}:=(u,P^{*}v)_{{\mathcal {S}}'(\mathbb {R} ^{n})\times {\mathcal {S}}(\mathbb {R} ^{n})}\,.}$

The same applies to pseudo differential operators . The operator adjoint with respect to the bilinear form is a pseudo differential operator and this can also be continued continuously analogous to an operator . Thereby is the space of the distributions and the space of the distributions with a compact carrier. ${\displaystyle P\colon {\mathcal {D}}(\mathbb {R} ^{n})\to {\mathcal {E}}(\mathbb {R} ^{n})}$ ${\displaystyle (\cdot ,\cdot )}$ ${\displaystyle P^{*}\colon {\mathcal {E}}(\mathbb {R} ^{n})\to {\mathcal {D}}(\mathbb {R} ^{n})}$ ${\displaystyle {\tilde {P}}\colon {\mathcal {E}}'(\mathbb {R} ^{n})\to {\mathcal {D}}'(\mathbb {R} ^{n})}$ ${\displaystyle {\mathcal {D}}'(\mathbb {R} ^{n})}$ ${\displaystyle {\mathcal {E}}'(\mathbb {R} ^{n})}$

Pseudo differential operators on manifolds

Let the space of the test functions be on , be a compact smooth manifold and be a map of . A steady map ${\displaystyle C_{c}^{\infty }(X)}$ ${\displaystyle X\subset \mathbb {R} ^{n}}$ ${\displaystyle M}$ ${\displaystyle (U_{j},\phi _{i})}$ ${\displaystyle M}$

${\displaystyle P\colon C^{\infty }(M)\to C^{\infty }(M)}$

is a pseudo differential operator if it can be represented locally in each map like a pseudo differential operator in . In concrete terms, this means, is a pseudo differential operator, if for with in a neighborhood of the operator ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle P}$ ${\displaystyle \psi _{0},\,\psi _{1}\in C_{c}^{\infty }(U_{j})}$ ${\displaystyle \psi _{1}=1}$ ${\displaystyle \operatorname {supp} (\psi _{0})}$

${\displaystyle {\tilde {P}}_{i}(u)(y):=\psi _{0}(x)\cdot P(\psi _{1}\cdot u\circ \phi _{i})(x)}$

with and is a pseudo differential operator. ${\displaystyle y=\phi _{i}(x)}$ ${\displaystyle u\in C^{\infty }(\phi _{i}(\Omega _{i}))}$

literature

• José García-Cuerva: Fourier Analysis and Partial Differential Equations. CRC Press, Boca Raton FL et al. 1995, ISBN 0-8493-7877-X .
• Lars Hörmander: The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators (= basic teachings of the mathematical sciences. Vol. 274). Springer, Berlin 1985, ISBN 3-540-13828-5 .
• Michail A. Shubin: Pseudodifferential Operators and Spectral Theory. 2nd edition. Springer, Berlin et al. 2001, ISBN 3-540-41195-X .
• Michael E. Taylor : Pseudodifferential Operators (= Princeton Mathematical Series. Vol. 34). Princeton University Press, Princeton NJ 1981, ISBN 0-691-08282-0 .
• Michael E. Taylor: Partial differential equations. Volume 1-2. Springer, New York et al. 1996, ISBN 0-387-94653-5 (Vol. 1), ISBN 0-387-94651-9 (Vol. 2).
• François Treves : Introduction to Pseudo Differential and Fourier Integral Operators. 2 volumes. Plenum Press, New York NY et al. 1980;
  • Volume 1: Pseudodifferential Operators. ISBN 0-306-40403-6 ;
  • Volume 2: Fourier Integral Operators. ISBN 0-306-40404-4 .

Web link

• Mark Joshi, Lectures on Pseudodifferential Operators, English

Individual evidence

1. ↑ Man Wah Wong: An introduction to pseudo-differential operator . 2nd Edition. World Scientific, River Edge NJ 1999, ISBN 981-02-3813-4 , pp. 31-33 .
2. ↑ Man Wah Wong: An introduction to pseudo-differential operator . 2nd Edition. World Scientific, River Edge NJ 1999, ISBN 981-02-3813-4 , pp. 54-60 .
3. ↑ Man Wah Wong: An introduction to pseudo-differential operator . 2nd Edition. World Scientific, River Edge NJ 1999, ISBN 981-02-3813-4 , pp. 62-69 .
4. ↑ Christopher D. Sogge: Fourier integral in Classical Analysis. (= Cambridge Tracts in Mathematics. Vol. 105). Digitally printed version. Cambridge University Press, Cambridge et al. 2008, ISBN 978-0-521-06097-4 , p. 106.