Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject. Besides the obvious fact that the Fourier transform is itself an oscillatory integral par excellence, one need only recall the occurrence of Bessel functions in the work of Fourier, the study of asymptotic related to these functions by Airy, Stokes, and Lipschitz, and Riemann’s use of the method of “stationary phase” in finding the asymptotic of certain Fourier transforms, all of which took place well over 100 years ago. The dyadic decomposition of a function Littlewood–Paley theory uses a decomposition of a function f into a sum of…