Oscillatory integral operator

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

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

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. $\lambda \,$ 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 $T_{\lambda }\,$ for large values of $\lambda \,$.

Oscillatory integral operators often appear in 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 E. Stein and his school.

Hörmander's theorem

The following bound on the $L^{2}\to L^{2}$ action of oscillatory integral operators (or $L^{2}$ operator norm) was obtained by L. Hörmander in his paper on Fourier integral operators:

Assume that $x,\,y\in \mathbb {R} ^{n}$, $n\geq 1$. Let $S(x,y)\,$ be real-valued and smooth, and let $a(x,y)\,$ be smooth and compactly supported. If $\mathop {\rm {det}} _{j,k}{\frac {\partial ^{2}S}{\partial x_{j}\partial y_{k}}}(x,y)\neq 0$ everywhere on the support of $a(x,y)\,$, then there is a constant $C\,$ such that $T_{\lambda }\,$, which is initially defined on smooth functions, extends to a continuous operator from $L^{2}(\mathbb {R} ^{n})\,$ to $L^{2}(\mathbb {R} ^{n})\,$, with the norm bounded by $C\lambda ^{-n/2}\,$, for any $\lambda \geq 1\,$:

$||T_{\lambda }||_{L^{2}(\mathbb {R} ^{n})\to L^{2}(\mathbb {R} ^{n})}\leq C\lambda ^{-n/2}.$