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 foundations of the theory derived by Lars Hörmander. Introduced in 1965 by Joseph Kohn she and Louis Nirenberg. Your study forms an important part of the micro- local analysis.
- 2.1 The Symbol Class
- 2.2 pseudo-differential operator
- 2.3 Actually worn pseudo- differential operator
- 3.1 Composition of pseudo-differential operators
- 3.2 adjoint operator
Linear differential operators with constant coefficients
Consider the linear differential operator with constant coefficients
Operating on the space of smooth functions with compact support in. It can as a composition of a Fourier transform, a simple multiplication by the polynomial ( the so-called symbol )
And the inverse Fourier transformation:
Be written. This is a multi- index, a differential operator, stands for derivative with respect to the - th component and are complex numbers.
Similarly, a pseudo- differential operator with symbol on an operator of the form
With a more general function in the integrand, as will be made further down.
The Fourier transform of a smooth function with compact support is in,
And inverse Fourier transform yields
Applying on this representation and use of
There is obtained ( 1).
Representation of solutions of partial differential equations
To a partial differential equation
Both sides are (formal) Fourier transforms to solve, with resulting algebraic equations:
If the symbol is always equal to zero for, can you by:
The solution is then treated with application of the inverse Fourier transformation:
Here, the following is assumed:
The last assumption can be weakened to the theory of distributions. The first two assumptions may be mitigated as follows:
You sit in the last formula is the Fourier transform of f a:
This is similar to formula (1 ), except that no polynomial, but a function of a more general nature
Definition of the pseudo- differential operator
The Symbol Class
Is an infinitely differentiable function on, open, with
For all, which is compact, for all and all multi- indices a constant and real numbers m, then a belongs to the symbol class.
Pseudo- differential operator
Again let a be a smooth function from the symbol class. A pseudo-differential operator of order m is usually a picture
Is defined. The space is the space of test functions, the space of smooth functions and is the Schwartz space.
Actually worn pseudo- differential operator
Be a pseudo- differential operator. In the following,
The integral kernel of the operator. The pseudo- differential operator is actually called supported if the projections are.
- Linear differential operators of order m with smooth bounded coefficients can be interpreted as a pseudo-differential operators of order m.
- The integral kernel
- The transpose of a pseudo- differential operator is also again a pseudo- differential operator.
- If a linear differential operator of order m is elliptic, its inverse is a pseudodifferential operator of order -m. So you can explicitly solve linear, elliptic differential equations more or less with the help of the theory of pseudo-differential operators.
- Differential operators are local. This means that one needs to know only the value of a function in the neighborhood 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 support of a distribution. It is therefore
- Since the Schwartz space is dense in the space of square integrable functions, it is possible by continuity arguments a pseudo-differential operator to continue. Also then applies a limited is so compact operator.
Composition of pseudo-differential operators
Pseudo-differential operators with the Schwartz space as a domain make this into itself. You are even an isomorphism. In addition, supported pseudo-differential operators actually make the space into itself. Therefore, it is possible for such operators, the composition of two pseudo-differential operators consider what again results in a pseudo-differential operators.
Let and be two symbols and are and the corresponding pseudo-differential operators, then is again a pseudo- differential operator. The symbol of the operator is an element of the space and it has the asymptotic expansion
Is for each pair of Schwartz functions
A bilinear form and be a pseudo- differential operator with symbol. Then the formal adjoint operator with respect to another pseudo differential operator and its symbol is an element of the space and it has the asymptotic expansion
Pseudo- differential operators on distribution spaces
Using the formal adjoint operator, it is possible to define pseudo- differential operators on distribution spaces. To this end, we consider instead of the bilinear dual pairing between the Schwartz space and its dual space. The dual pairing can be understood as a continuous extension of. Therefore, it is possible pseudo-differential operators to define the space of tempered distributions, ie on the dual space of the Schwartz space.
Be a pseudo- differential operator and a tempered distribution. Then the continuous operator for all defined by
For pseudo-differential operators analogous situation applies. The bilinear form with respect to the adjoint operator is a pseudo-differential operator, and this one can also pursue similar to an operator continuously. This is the space of distributions and the space of distributions with compact support.
Pseudo-differential operators on manifolds
Be the space of test functions on, be a compact smooth manifold and is a map of. A continuous mapping
Is a Pseudodifferentialopertor, if it can be represented locally in each map as a pseudo- differential operator in. Specifically, this is a pseudo- differential operator, if for with in a neighborhood of the operator
And with a pseudo-differential operator.