You might remember that I contributed some lecture notes on Littlewood-Paley theory to a masterclass, which I then turned into a series of three posts (I – II – III). I have also contributed a lecture note on some basic theory of oscillatory integrals, and therefore I am going to do the same and share them here as a blog post in two parts. The presentation largely follows the one in Stein’s “Harmonic Analysis: Real-variable methods, orthogonality, and oscillatory integrals“, with inputs from Stein and Shakarchi’s “Functional Analysis: Introduction to Further Topics in Analysis“, from some lecture notes by Terry Tao for his 247B course, from a very interesting paper by Carbery, Christ and Wright and from a number of other places that I would now have trouble tracking down.
In this first part we will discuss the theory of oscillatory integrals when the phase is a function of a single variable. There are extensive exercises included that are to be considered part of the lecture notes; indeed, in order to keep the actual notes short and engage the reader, I have turned many things into exercises. If you are interested in learning about oscillatory integrals, you should not ignore them.
In the next post, we will study instead the case where the phases are functions of several variables.
A large part of modern harmonic analysis is concerned with understanding cancellation phenomena happening between different contributions to a sum or integral. Loosely speaking, we want to know how much better we can do than if we had taken absolute values everywhere. A prototypical example of this is the oscillatory integral of the form
Here , called the amplitude, is usually understood to be “slowly varying” with respect to the real-valued , called the phase, where denotes a parameter or list of parameters and gets larger as grows; for example . Thus the oscillatory behaviour is given mainly by the complex exponential .
Expressions of this form arise quite naturally in several problems, as we will see in Section 1, and typically one seeks to provide an upperbound on the absolute value of the integral above in terms of the parameters . Intuitively, as gets larger the phase changes faster and therefore oscillates faster, producing more cancellation between the contributions of different intervals to the integral. We expect then the integral to decay as grows larger, and usually seek upperbounds of the form . Notice that if you take absolute values inside the integral above you just obtain , a bound that does not decay in at all.
The main tool we will use is simply integration by parts. In the exercises you will also use a little basic complex analysis to obtain more precise information on certain special oscillatory integrals.
In this section we shall showcase the appearance of oscillatory integrals in analysis with a couple of examples. The reader can find other interesting examples in the exercises.
1.1. Fourier transform of radial functions
Let be a radially symmetric function, that is there exists a function such that for every . Let’s suppose for simplicity that (equivalently, that ), so that it has a well-defined Fourier transform. It is easy to see (by composing with a rotation and using a change of variable in the integral defining ) that must also be radially symmetric, that is there must exist such that ; we want to understand its relationship with . Therefore we write using polar coordinates
where denotes the surface measure on the unit -dimensional sphere induced by the Lebesgue measure on the ambient space . By inspection, we see that the integral in brackets above is radially symmetric in , and so if we define
This is the relationship we were looking for: it allows one to calculate the Fourier transform of directly from the radial information .