Oscillatory integral

Summary

An oscillatory integral is an integral of the form $\int e^{i\lambda\phi(x)} \psi(x) \, dx$. Here $\phi$ is a real phase function, $\psi$ is an amplitude and $\lambda$ is a large real parameter. The main question is how fast it decays as $\lambda \to \infty$, which depends on the critical points of $\phi$. Fourier transforms of measures on curved surfaces, Bessel functions and solutions of dispersive equations are all oscillatory integrals. Their estimates are basic tools in harmonic analysis, number theory and PDE.