This is the second and final part of an entry dedicated to a very interesting and inventive trick due to Bourgain. In part I we saw a lemma on maximal Fourier projections due to Bourgain, together with the context it arises from (the study of pointwise ergodic theorems for polynomial sequences); we also saw a baby version of the idea to come, that we used to prove the Rademacher-Menshov theorem (recall that the idea was to represent the indices in the supremum in their binary positional notation form and to rearrange the supremum accordingly). Today we finally get to see Bourgain’s trick.
Before we start, recall the statement of Bourgain’s lemma:
Lemma 1 [Bourgain]: Let be an integer and let a set of distinct frequencies. Define the maximal frequency projections
where the supremum is restricted to those with being the smallest integer such that .
Here we are using the notation in the statement in place of the expanded formula . Observe that by the definition of we have that the intervals are disjoint (and is precisely maximal with respect to this condition).
We will need to do some reductions before we can get to the point where the trick makes its appearance. These reductions are the subject of the next section.
3. Initial reductions
A first important reduction is that we can safely replace the characteristic functions by smooth bump functions with comparable support. Indeed, this is the result of a very standard square-function argument which was already essentially presented in Exercise 22 of the 3rd post on basic Littlewood-Paley theory. Briefly then, let be a Schwartz function such that is a smooth bump function compactly supported in the interval and such that on the interval . Let (so that ) and let for convenience denote the difference . We have that the difference
is an bounded operator with norm (that is, independent of ). Indeed, observe that , and bounding the supremum by the sum we have that the norm (squared) of the operator above is bounded by
where the summation in is restricted in the same way as the supremum is in the lemma (that is, the intervals must be pairwise disjoint). By an application of Plancherel we see that the above is equal to
but notice that the functions have supports disjoint in , and therefore the multiplier satisfies in a neighbourhood of , and vanishes outside such neighbourhood. A final application of Plancherel allows us to conclude that the above is bounded by by orthogonality (these neighbourhoods being all disjoint as well).
By triangle inequality, we see therefore that in order to prove Lemma 1 it suffices to prove that the operator
is bounded with norm at most .