In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases.
A Fourier integral operator is given by:
where denotes the Fourier transform of , is a standard symbol which is compactly supported in and is real valued and homogeneous of degree in . It is also necessary to require that on the support of a. Under these conditions, if a is of order zero, it is possible to show that defines a bounded operator from to .[1]
^Hörmander, Lars (1970), "Fourier integral operators. I", Acta Mathematica, 127, Springer Netherlands: 79–183, doi:10.1007/BF02392052
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