Oscillatory integral operator

In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form

${\displaystyle T_{\lambda }u(x)=\int _{\mathbb {R} ^{n}}e^{i\lambda S(x,y)}a(x,y)u(y)\,dy,\qquad x\in \mathbb {R} ^{m},\quad y\in \mathbb {R} ^{n},}$

where the function S(x,y) is called the phase of the operator and the function a(x,y) is called the symbol of the operator. λ is a parameter. One often considers S(x,y) to be real-valued and smooth, and a(x,y) smooth and compactly supported. Usually one is interested in the behavior of T_λ for large values of λ.

Oscillatory integral operators often appear in many fields of mathematics (analysis, partial differential equations, integral geometry, number theory) and in physics. Properties of oscillatory integral operators have been studied by Elias Stein and his school.^[1]