History
The study of pseudo-differential operators began in the mid 1960s with the work ofMotivation
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: Here, is a multi-index, are complex numbers, and : is an iterated partial derivative, where ∂''j'' means differentiation with respect to the ''j''-th variable. We introduce the constants to facilitate the calculation of Fourier transforms. ;Derivation of formula () The Fourier transform of a smooth function ''u'',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 ofDefinition 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'': 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 pseudo-differential operator of order m and belongs to the classProperties
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 (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 aKernel of pseudo-differential operator
Pseudo-differential 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
*Footnotes
References
* . *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. * M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag 2001. * Francois Treves, Introduction to Pseudo Differential and Fourier Integral Operators, (University Series in Mathematics), Plenum Publ. Co. 1981. * F. G. Friedlander and M. Joshi, Introduction to the Theory of Distributions, Cambridge University Press 1999. * * André Unterberger, ''Pseudo-differential operators and applications: an introduction''. Lecture Notes Series, 46. Aarhus Universitet, Matematisk Institut, Aarhus, 1976.External links