When faced with the task of bounding an integral such as
where is "oscillatory" (e.g. for some rapidly oscillating ) and is "smooth" (so that is "small"), then it is often advantageous to use integration by parts to integrate and differentiate . In some cases one may wish to transfer a term from the smooth factor to the oscillatory factor to make easier to integrate (e.g. borrow a factor of from to turn into , which has an antiderivative of ).
Fourier transform of bump functions or Schwartz functions is rapidly decreasing. Consider a Schwartz function , and denote by its Fourier transform. We will use integration by parts and the fact that together with its derivatives of any order decay faster than any polynomial function at infinity to exhibit the rapid decay of at infinity. Using the definition of and 'borrowing' a factor of from the function we write
Using the same integration by parts trick on the higher order derivatives of it is not difficult to see that
for any . Now that the integration by parts has exploited the cancellation due to the oscillating factor we can put absolute values everywhere and estimate
Of course this estimate makes sense since is a Schwartz function and thus for any .
A slightly more general statement is the following. Let be a polynomial and write for the differential operator . We consider the differential operator , defined in the obvious way. Using the special case that we consider before and the linearity of the Fourier transform we can now write the identity
for any , which in turn implies the estimate
Deducing the van der Corput lemma for oscillatory integrals in the simplest case. Suppose that the phase function obeys the bound for all . Assume also that is monotone. We want to prove the bound
for . Here and but more general functions can be considered as in Example 3 below. 'Borrowing' a factor of from the function we can write . Now we use integration by parts to get
Now we have exploited the cancellation that comes from the oscillating factor so we can estimate more or less by brute force
In order to complete the estimate we note that has constant sign throughout since is monotone and thus
Deducing van der Corput for bump functions from the van der Corput lemma for oscillatory integrals. We now consider the slightly more general case than the one of Example 2 where and is a smooth bump function on . We will deduce van der Corput's lemma for bump functions, that is the estimate
where obeys the bound for , assuming we know the van der Corput lemma for oscillatory integrals. Using integration by parts we write
The basic point here is that van der Corput's estimate is uniform over choices of intervals where the hypothesis holds. Thus it makes sense to estimate
Now the term is controlled by the van der Corput lemma for oscillatory integrals by so we get the desired estimate.
Here we give an example of the localization principle in the method of stationary phase (also known as the "principle of non-stationary phase"): if is a bump function and is a smooth phase which is never stationary on the support of (thus is bounded away from zero on this support), then is rapidly decreasing in .
By taking absolute values everywhere (or by the base times height bound) we see that this integral is already bounded in . To get one order of decay, one can rewrite the integral (using the multiplying and dividing trick), taking advantage of the non-vanishing of , as
The point is that the second factor is the derivative of . Integrating by parts, we can express this as
Applying the base times height bound now gives a bound of , giving one order of decay. Repeating this process, one can show arbitrary amounts of decay in (note that while many derivatives of will begin appearing, only the first derivative will appear in the denominator of the integrand).
This technique works well for establishing energy identities or energy inequalities in PDE. We illustrate this with the Korteweg-de Vries equation
where is a function, which for simplicity we will assume to be smooth, with all derivatives rapidly decreasing as , in order to ignore issues about justifying integration by parts, differentiation under the integral sign, etc.. We claim that the mass
is conserved. To see this, it of course suffices to show that vanishes. Differentiating under the integral sign, we obtain
where we have suppressed the independent variables for brevity. Using (1), this becomes
To simplify this using integration by parts, we first observe that is a total derivative , and so by integration by parts (or the fundamental theorem of calculus) the second term vanishes.
What about the first term ? The problem here is that one factor, , is absorbing too many of the derivatives. But we can rebalance the derivatives by integration by parts, writing
(Intuitively, has a lot of cancellation in it, while has a lot of regularity (at least relative to , and integration by parts is the perfect tool to rebalance the two.) But now we observe that is also a total derivative , so vanishes as required.
The technique also works for improper integrals such as provided there is enough decay at infinity. In fact, the method tends to work even better in this case, as one does not pick up boundary terms.
In some cases, the method can be iterated, integrating an oscillation repeatedly and differentiating all other factors. This often leads to bounds which are rapidly decreasing in some key parameter that controls the amount of oscillation.
The trick also works in higher dimensions; one picks a direction (or vector field) to integrate one factor in, and differentiates the other (using Stokes' theorem if necessary).
See also partial summation for a discrete version of this trick.