Not long ago we discussed one of the main direct applications of the Littlewood-Paley theory, namely the Marcinkiewicz multiplier theorem. Recall that the single-variable version of this theorem can be formulated as follows:
- for every Littlewood-Paley dyadic interval with
where denotes the total variation of over the interval .
Then for any the multiplier defined by for functions extends to an bounded operator,
You should also recall that the total variation above is defined as
and if is absolutely continuous then exists as a measurable function and the total variation over interval is given equivalently by . We have seen that the “dyadic total variation condition” 2.) above is to be seen as a generalisation of the pointwise condition , which in dimension 1 happens to coincide with the classical differential Hörmander condition (in higher dimensions the pointwise Marcinkiewicz conditions are of product type, while the pointwise Hörmander(-Mihklin) conditions are of radial type; see the relevant post). Thus the Marcinkiewicz multiplier theorem in dimension 1 can deal with multipliers whose symbol is somewhat rougher than being differentiable. It is an interesting question to wonder how much rougher the symbols can get while still preserving their mapping properties (or maybe giving up some range – recall though that the range of boundedness for multipliers must be symmetric around 2 because multipliers are self-adjoint).
Coifman, Rubio de Francia and Semmes came up with an answer to this question that is very interesting. They generalise the Marcinkiewicz multiplier theorem (in dimension 1) to multipliers that have bounded -variation with > 1. Let us define this quantity rigorously.
Definition: Let and let be an interval. Given a function , its -variation over the interval is
Notice that, with respect to the notation above, we have . From the fact that when we see that we have always , and therefore the higher the the less stringent the condition of having bounded -variation becomes (this is linked to the Hölder regularity of the function getting worse). In particular, if we wanted to weaken hypothesis 2.) in the Marcinkiewicz multiplier theorem above, we could simply replace it with the condition that for any Littlewood-Paley dyadic interval we have instead . This is indeed what Coifman, Rubio de Francia and Semmes do, and they were able to show the following:
- for every Littlewood-Paley dyadic interval with
Then for any such that the multiplier defined by extends to an bounded operator,
The statement is essentially the same as before, except that now we are imposing control of the -variation instead and as a consequence we have the restriction that our Lebesgue exponent satisfy . Taking a closer look at this condition, we see that when the variation parameter is the condition is empty, that is there is no restriction on the range of boundedness of : it is still the full range < < , and as grows larger and larger the range of boundedness restricts itself to be smaller and smaller around the exponent (for which the multiplier is always necessarily bounded, by Plancherel). This is a very interesting behaviour, which points to the fact that there is a certain dichotomy between variation in the range below 2 and the range above 2, with -variation being the critical case. This is not an isolated case: for example, the Variation Norm Carleson theorem is false for -variation with ; similarly, the Lépingle inequality is false for 2-variation and below (and this is related to the properties of Brownian motion).
Today, as a natural continuation to the posts on Littlewood-Paley theory and its applications, I am going to present the interesting proof of this very nice theorem. I found it impossible to get a hold of the original paper without taking a trip to the library, so being lazy I am going to follow instead Lacey’s nice presentation (which I suppose is very close to the original one anyway). The reasons why I believe the proof to be interesting are several: for starters, the proof is surprisingly simple; moreover, it relies on a generalisation of the Littlewood-Paley theorem due to Rubio de Francia which is worth seeing once you have studied the basic theory of square functions; finally, one very important ingredient in the proof is a partition of dependent on the symbol which is very much alike that beautiful idea which is used to prove the maximal Hausdorff-Young inequality of Christ-Kiselev.
To summarise the proof before we start: we will build sub-partitions of the partition of given by the Littlewood-Paley dyadic intervals, in such a manner that on each subpartition we have a uniform control of the norm of (part of) the multiplier symbol ; then we will use square functions adapted to these partitions to control the resulting multipliers, somewhat analogously to what is done in the Marcinkiewicz case (but conceptually even simpler). Finally we will combine everything together to conclude. In the next section we will present the necessary inequalities of Rubio de Francia (without proof) before presenting the proof.
1. Rubio de Francia square functions
Recall that the Littlewood-Paley theorem says the following: if denotes the frequency projection given by and are the Littlewood-Paley dyadic intervals, the collection of which we denote by , then we have for any
The first person to go beyond this statement was Lennart Carleson, who investigated the square function
As you can see, the intervals on which we are taking the frequency projections are no longer dyadic – rather, they all have the same (unit) length. If you recall, the heuristic motivation behind the Littlewood-Paley theory is that, since and the different frequency pieces are dyadically separated, we should expect a random cancellation between the different terms of the frequency decomposition – which means that most of the time the sum should have magnitude approximately . This heuristic no longer works in the case where the frequency intervals are the ‘s, because there are now many many terms in the frequency decomposition that have comparable frequencies and will therefore be “aligned” for longer periods of time, or “correlated” if you prefer – there will not be much cancellation between them. This leads us to suspect that the analogue of the Littlewood-Paley theorem for Carleson’s square function should fail, at least partially. This is indeed the case. In fact, Carleson showed that the inequality is FALSE when < 2. This is actually very simple to check: just take such that for a large integer ( is a continuous version of the Dirichlet kernel). It is easy to estimate that (the contribution from the peak around the origin dominates). As for the square function, we simply have
and therefore the inequality can only be true if for any large , which in turn is only possible if . Carleson then went on to prove that in this range the inequality is actually true. His paper, titled “On the Littlewood-Paley theorem” and published in 1967 on the Report of the Mittag-Leffler Institute seems to have vanished – not even MathSciNet has it in their records. Indeed, I think the result went largely unnoticed, since more than a decade later (1981) Córdoba re-proved the theorem in his work on Bochner-Riesz multipliers.
Nevertheless, this work was later subsumed by work of Rubio de Francia, who generalised it to arbitrary collections of disjoint intervals. Indeed, he proved the following:
Theorem 3 [Rubio de Francia, ’85]: Let be a collection of disjoint intervals and let denote the associated square function
Importantly, the constant is independent of the collection .
Some remarks are in order:
- The sharp reader will have already noticed that, since the intervals are disjoint, the case of Theorem 3 is actually a trivial consequence of Plancherel.
- As seen in the proof of the Littlewood-Paley theorem, if the collection partitions then we have by duality and Cauchy-Schwarz that inequality (1) for exponent has the consequence that
where now . Thus we can see Rubio de Francia's result as saying that the heuristic above still applies to arbitrary partitions of , but only in the range – the heuristic being that most of the time .
- Of course, depending on the collection of intervals, the condition might not be sharp, as is the case when we take the Littlewood-Paley dyadic intervals and can thus go below 2. It is not currently known (despite the result being nearly 40 years old) when the condition is sharp, that is, there is no characterisation of the collections for which the theorem fails in the range < 2. It is conjectured that the condition is sharp essentially for all collections that are not "lacunary" in some sense, though it is even hard to understand what the correct notion of lacunarity to state a conjecture should be.
There are a number of proofs of Rubio de Francia’s theorem, though we will not see a single one in here. Rubio de Francia’s original proof worked roughly as follows: first of all, using classical Littlewood-Paley theory and Whitney decompositions of the intervals, one reduces to the case where the intervals are all well-separated, meaning that if are disjoint then are also disjoint; then, the theorem is reduced to proving the same statement for a square function with the same frequency information but smoother frequency projections (much like in the proof of the Littlewood-Paley theorem); finally, the boundedness of is deduced by interpolation between the trivial case and the endpoint inequality – which is not hard to prove as a consequence of some simple vector-valued kernel estimates.
Bourgain reproved the theorem by proving the endpoint of the dual inequality, thus extending the result somewhat; that is, he proved that when is a partition of the real line then
where denotes the quasi-norm of the (real) Hardy space . Bourgain’s paper “On square functions on the trigonometric system” is very hard to get, having been published in a journal that does not exist in that form anymore. The proof he gives is beautiful (not a surprise) and I will probably talk about it in the future.
Another vividly distinct proof of Theorem 3 is given by Lacey in the aforementioned notes, in which the theorem is reformulated in time-frequency language (using wavepackets) and a time-frequency proof is given, with plenty of details. These notes include a discussion of many issues related to the Rubio de Francia inequalities and are a must-read for whoever is interested in such inequalities.
Finally, there is also a not-yet-published paper of Benea and Muscalu in which they re-prove Rubio de Francia’s theorem in yet another time-frequency way (distinct from Lacey’s).
Other inputs, variations and extensions to higher dimensions have been given by Soria, Sjölin, Journé, Sato, Zhu and maybe others I am forgetting at the moment; I will not discuss these here.
2. Proof of Theorem 2
With the Rubio de Francia inequalities at hand, we are ready to prove Theorem 2.
Proof: The idea is to reduce to the simpler case of multipliers which are simple functions on each Littlewood-Paley dyadic interval. In particular, assume that is of the form , where each collection is a partition of the Littlewood-Paley interval and is a complex coefficient. Assume furthermore that these subpartitions of Littlewood-Paley intervals are bounded in cardinality, that is assume that for every we have . Then we can argue as follows, following the footsteps of the proof of the Marcinkiewicz multiplier theorem (Theorem 1 above): if we let denote the Littlewood-Paley square function, we have by Littlewood-Paley theorem that
Due to the simple form the symbol has, we see that
having an sum is inconvenient in this context, and therefore we apply Cauchy-Schwarz to the latter to get a nice square function instead,
Performing the -summation in we see that we have shown
where is the collection of all intervals, that is . Now the object on the RHS is a Rubio de Francia square function, which we know is bounded at least when . Assume therefore that this is the case (which is not a limitation, because multipliers have range of boundedness symmetric about exponent 2), and as a consequence of (1) we have therefore that
for all . This bound is not so great because there is a large loss in the parameter , but at this point we should observe something: when this factor is not there! Indeed, in that case we have simply by Plancherel; but this means that for any we can (complex) interpolate between all these estimates and lower the exponent somewhat. Indeed, for a fixed exponent , we can write as an interpolation exponent between 2 and any extremely large exponent ; in practice, the result will be the same as if we had assumed (once we take a limit), although obviously we are not allowed to use precisely this exponent. The result is the following: if is such that
for all . We have improved the constant a little! This small improvement will go a long way though.
Now that we have some partial result, how can we exploit it? Can we reduce the multiplier symbol in Theorem 2 to a symbol of the type just considered above? It turns out that it is not at all hard to reduce the symbol to a sum of symbols of the above form, in such a way that (2) will give a summable contribution. The argument is an ingenious decomposition of in martingale differences where the martingale is dictated from itself (by its -variation, precisely). Let us see how.
We assume for simplicity that the constant is 1, that is we normalise the symbol so that for any Littlewood-Paley interval we have . Fix such an and let ; we want to partition into intervals that carry uniform “-variation-mass”, so to speak. This is easy to achieve in the following way: let denote the function
that is, is the -variation of in the interval from the left endpoint of to the point , raised to the power (so that we have additivity). Function is clearly a monotone increasing function, and therefore has a well-defined inverse function. Split therefore the interval into equal disjoint intervals, that is,
with ; we define then for any
which is an interval, by the monotonicity of . With this definition, we have by construction.
Define the collection to be the collection of all the intervals resulting from this procedure:
we have that:
- each collection is a partition of ;
- each collection is a refinement1 of the Littlewood-Paley partition and each is partitioned by into at most intervals;
- the collections are refinements of each other: if then is a refinement of . Notice that each interval of is split into (at most) 2 subintervals by .
Observe that property 2.) above points in the direction of (2), in that we have uniform control on the cardinality of the (sub-)partitions, but we still have to decompose the symbol. Using properties 1.) and 3.), we do a martingale decomposition of adapted to these collections: let (the sigma-algebra generated by the collection ) and notice that is an increasing sequence of sigma-algebras; therefore, if we introduce the martingale differences
we can decompose every function into
Now it is worth observing what happens when we apply the martingale decomposition to itself. Consider a fixed and a fixed and observe that
where is the unique interval in such that and is the unique interval in that contains . Observe that there is a unique affine map that maps to while preserving the ordering; using this map as a change of variables we can thus write
Notice though that for any we have that trivially
; but by construction this quantity is controlled by ! Therefore, since the above expression is an average, we have the pointwise bound
Combining this bound with property 2.) of the collections we see that for the symbols we have by inequality (2) that
If the overall exponent of above is negative and therefore the quantity is summable; by triangle inequality we have therefore that
provided the condition is satisfied (the term comes from bounding using the standard Littlewood-Paley square function). This concludes the nice proof of Theorem 2.
We close this post with a remark and a question:
Remark: If we had not improved the exponent of in inequality (2), we would have only concluded that . This is still summable in when , so we would still have concluded something, but we would have missed the part of Theorem 2.
A question: Is the range 2, can we find a symbol with finite but such that the multiplier is unbounded for such that ?
Presumably the answer is yes, but I have not thought about the answer, honestly.