Fourier integral operator

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 ${\displaystyle T}$ is given by:

${\displaystyle (Tf)(x)=\int _{\mathbb {R} ^{n}}e^{2\pi i\Phi (x,\xi )}a(x,\xi ){\hat {f}}(\xi )\,d\xi }$

where ${\displaystyle {\hat {f}}}$ denotes the Fourier transform of ${\displaystyle f}$, ${\displaystyle a(x,\xi )}$ is a standard symbol which is compactly supported in ${\displaystyle x}$ and ${\displaystyle \Phi }$ is real valued and homogeneous of degree ${\displaystyle 1}$ in ${\displaystyle \xi }$. It is also necessary to require that ${\displaystyle \det \left({\frac {\partial ^{2}\Phi }{\partial x_{i}\,\partial \xi _{j}}}\right)\neq 0}$ on the support of a. Under these conditions, if a is of order zero, it is possible to show that ${\displaystyle T}$ defines a bounded operator from ${\displaystyle L^{2}}$ to ${\displaystyle L^{2}}$.^[1]