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