I have written some notes on Littlewood-Paley theory for a masterclass, which I thought I would share here as well. This is the first part, covering some motivation, the case of a single frequency projection and its vector-valued generalisation. References I have used in preparing these notes include Stein’s “*Singular integrals and differentiability properties of functions*“, Duoandikoetxea’s “*Fourier Analysis*“, Grafakos’ “*Classical Fourier Analysis*” and as usual some material by Tao, both from his blog and the notes for his courses. Prerequisites are some basic Fourier transform theory, Calderón-Zygmund theory of euclidean singular integrals and its vector-valued generalisation (to Hilbert spaces, we won’t need Banach spaces).

**0. Introduction**

Harmonic analysis makes a fundamental use of divide-et-impera approaches. A particularly fruitful one is the decomposition of a function in terms of the frequencies that compose it, which is prominently incarnated in the theory of the Fourier transform and Fourier series. In many applications however it is not necessary or even useful to resolve the function at the level of single frequencies and it suffices instead to consider how wildly different frequency components behave instead. One example of this is the (formal) decomposition of functions of given by

where denotes the operator

commonly referred to as a (dyadic) *frequency projection*. Thus represents the portion of with frequencies of magnitude . The Fourier inversion formula can be used to justify the above decomposition if, for example, . Heuristically, since any two oscillate at significantly different frequencies when is large, we would expect that for most ‘s the different contributions to the sum cancel out more or less randomly; a probabilistic argument typical of random walks (see Exercise 1) leads to the conjecture that should behave “most of the time” like (the last expression is an example of a *square function*). While this is not true in a pointwise sense, we will see in these notes that the two are indeed interchangeable from the point of view of -norms: more precisely, we will show that for any it holds that

This is a result historically due to Littlewood and Paley, which explains the name given to the related theory. It is easy to see that the case is obvious thanks to Plancherel’s theorem, to which the statement is essentially equivalent. Therefore one could interpret the above as a substitute for Plancherel’s theorem in generic spaces when .

In developing a framework that allows to prove () we will encounter some variants of the square function above, including ones with smoother frequency projections that are useful in a variety of contexts. We will moreover show some applications of the above fact and its variants. One of these applications will be a proof of the boundedness of the spherical maximal function (almost verbatim the one on Tao’s blog).

**Notation:** We will use to denote the estimate where is some absolute constant, and to denote the fact that . If the constant depends on a list of parameters we will write .

**1. Motivation: estimates for the heat equation with sources**

In this section we present certain natural objects that cannot be dealt with by (euclidean) Calderón-Zygmund theory alone, as a way to motivate the study of Littlewood-Paley theory.

with representing the instantaneous heat generated by the source at point at instant . If the total heat introduced by the sources is small, we expect to see that the solution evolves slowly (that is, is also small). One sense in which this conjecture can be made rigorous is to control the -norms of by the corresponding norms of the data . If we take a Fourier transform in both and we have

Now, we are trying to relate and on the Fourier side (that is, we are trying to use spectral calculus). The simplest way to do so is to observe that, by taking the Fourier transform of both sides of (2), we have

thus we can write with a little algebra

Let , so that the result can be restated as . Observe that both the real and imaginary parts of are discontinuous at and nowhere else. By Fourier inversion we can interpret this equality as defining a linear operator , given by

thus we have come to realise that the partial derivative can be obtained (at least formally) from the function by . To bound is therefore equivalent to bounding : if is bounded from into then we will have

which is the type of control we are looking for.

Observe that by the properties of the Fourier transform our operator is actually a convolution operator: precisely^{1},

If we knew that the convolution kernel satisfied we would be done, since we could appeal to Young’s inequality. Unfortunately, this is not the case. This is the same issue that one encounters in dealing with singular integral kernels such as , so the natural thing to do is to check whether is a Calderón-Zygmund kernel. However, this is also not the case! Indeed, for to be a (euclidean) Calderón-Zygmund convolution kernel, it has to satisfy certain smoothness properties such as . However, enjoys a certain (parabolic) scaling invariance that makes it incompatible with the last inequality: indeed, one can easily see that for any

and deduce from this a similar anisotropic rescaling invariance for . Since is not invariant with respect to this rescaling, the inequality cannot possibly hold for all .

One solution to this issue is to develop a theory of parabolic singular integrals that extends the results of classical Calderón-Zygmund theory to those kernels that satisfy anisotropic rescaling invariance identities of the kind above. While this is a viable approach, we will take a different one in these notes. The approach we will take will not make use of said identities and will thus be more general in nature. With the methods of next section we will be able to show that is indeed bounded for all .

**2. Frequency projections **

We begin by considering the frequency projections introduced in the beginning. Let be an interval and define the frequency projection to be

An equivalent way of defining this is to work directly on the frequency side of things and stipulate that is the operator that satisfies

this is well-defined for and can be therefore extended to functions . We claim that defines an bounded operator for any and any (possibly semi-infinite) interval and, importantly, that its norm is bounded independently of . The point is that the operator is essentially a linear combination of two (modulated) Hilbert transforms, that are bounded in the same stated range. Indeed, observe that the Fourier transform of the Hilbert transform kernel is ; it is easy to show that, if , then

Now, the Fourier transform exchanges translations with modulations, and specifically ; we see therefore that we can write, with the Hilbert transform,

If we let denote the modulation operator , we have therefore shown that

and from the boundedness of it follows that

(since does not change the norms at all).

At this point, another important thing to notice is that when are *disjoint* intervals, then the operators are orthogonal to each other. Indeed, one has by Parseval’s identity

A consequence of this fact, together with Plancherel’s theorem, is the case of the result that we mentioned in the introduction, that is inequality (); and actually, one has more, namely that for any collection of disjoint intervals that partition one has

Prove this in Exercise 2 to familiarise yourself with frequency projections.

** 3. Vector-valued frequency projections **

As a step towards proving () (and also important for the applications), it will be useful to study vector-valued analogues of these inequalities, which are easier to prove in general. We change the setting to to work in more generality: so, if is an axis-parallel rectangle (that is, of the form with intervals, possibly semi-infinite) in , we define the frequency projection to be the operator that satisfies for any

The following is an abstract vector-valued analogue of (3).

Proposition 1 (Vector-valued square functions)Let be a measure space with a -finite measure, and let denote the Hilbert space , that is the space of functions such that . We let be a measurable collection of arbitrary rectangles of (that is, the mapping is a measurable function^{2}).Given any vector-valued function (that is, a function of that takes values in ), we can define its vector-valued frequency projection by

Then we have that for any it holds for all functions that

In particular, the constant is independent of the collection of rectangles and even of the measure space .

The level of abstraction in the previous statement is quite high, and it might take the reader a while to fully unpack its meaning. The following special case of Proposition 1 should help:

Corollary 2Let be a collection of arbitrary axis-parallel rectangles in . If is a sequence of functions such that is in , we have

with constant independent of the collection of rectangles.

To prove () we will only need Corollary 2, but for the application of Littlewood-Paley theory we will give in a subsequent post (that is, the Marcinkiewicz multiplier theorem) we will need the full power of Proposition 1. Prove in Exercise 3 that the corollary is indeed just a special case of the proposition, before looking at the proof below.

Remark 1It is important to notice that there is no assumption on the family of rectangles other than the fact that they are all axis-parallel – in particular, they are not necessarily pairwise disjoint or even distinct.

*Proof:* The proof is a vector-valued generalisation of what we said for frequency projections over intervals in the previous section.

First consider and introduce the vector-valued Hilbert transform given by

we claim that this operator is bounded. Indeed, is given by convolution with the -valued^{3} kernel , and this kernel satisfies (as you can easily verify)

- ,
- ,
- for any .

Then i)-ii)-iii) imply boundedness simply by vector-valued Calderón-Zygmund theory.

Next, observe that if and the intervals are arbitrary, we can use the same trick used to prove (3) (rewriting as sum of Hilbert transforms conjugated with a modulation) to deduce boundedness of the corresponding vector-valued operator from the boundedness of above.

Finally, in the case of general one should observe that if then factorises as^{4}

where denotes frequency projection in the variable. Therefore, writing and , we have by the one-dimensional result applied to that

iterating the argument for each variable, we obtain the claimed boundedness for generic .

In the next part of the notes we will consider smooth frequency projections and finally prove (). We will also see how the theory generalises to higher dimensions.

** EXERCISES **

Exercise 1Let be independent random variables taking values in such that takes each value with equal probability. Show that

Generalise this to the case where the sum is with some complex coefficients.

Exercise 2Show that (4) holds for any collection of disjoint intervals.

Exercise 3Show that Corollary 2 is a special case of Proposition 1. What is the measure space that gives the corollary? Go to the proof of the proposition and spell out all the Hilbert norms in the argument in terms of the measure space you found.

**Footnotes:**

^{1}: With a slight abuse of notation we have used to denote the inverse Fourier transform as well, in place of the more usual , motivated by the fact that on the two differ only by a reflection. We will continue to do so in the rest of these notes as it will always be clear from context which one is meant.

^{2}: Notice the space of rectangles of can be identified with .

^{3}: is the space of bounded linear operators in . Specifically, is the operator that multiplies componentwise by .

^{4}: The identity can be proved by appealing to the fact that it is trivial for tensor products and tensor products of single variable functions are dense in .