I found these fantastic lecture notes on Penrose’s aperiodic tilings by Alexander F. Ritter, which he wrote for a masterclass at Oxford in 2014.

Lectures on Penrose tilings [Mirror]

There is not much systematic well-structured material on aperiodic tilings around, so I thought I would share this for whoever is interested. The material is accessible to every undergrad that has taken a class in real analysis (some nice application of extracting a converging subsequence from a sequence in a compact space, in the proof of the Extension Theorem), and includes many painstaking hand-drawings that are very helpful in following the arguments. Besides being overall well-written, the notes are also good at hammering home some important points – for example, the fact that the Composition and Decomposition operations that one can make on a tiling are unique and thus reversible in the case of tilings by Penrose’s Kites and Darts tiles, which has a cascade of consequences (they prove aperiodicity of the tilings in a few lines, for example) which do not apply to regular periodic tilings for this very reason (non-uniqueness).

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Carbery's proof of the Stein-Tomas theorem

Writing the article on Bourgain’s proof of the spherical maximal function theorem I suddenly recalled another interesting proof that uses a trick very similar to that of Bourgain – and apparently directly inspired from it. Recall that the “trick” consists of the following fact: if we consider only characteristic functions as our inputs, then we can split the operator in two, estimate these parts each in a different Lebesgue space, and at the end we can combine the estimates into an estimate in a single space by optimising in some parameter. The end result looks as if we had done “interpolation”, except that we are “interpolating” between distinct estimates for distinct operators!

The proof I am going to talk about today is a very simple proof given by Tony Carbery of the well-known Stein-Tomas restriction theorem. The reason I want to present it is that I think it is nice to see different incarnations of a single idea, especially if applied to very distinct situations. I will not spend much time discussing restriction because there is plenty of material available on the subject and I want to concentrate on the idea alone. If you are already familiar with the Stein-Tomas theorem you will certainly appreciate Carbery’s proof.

As you might recall, the Stein-Tomas theorem says that if denotes the Fourier restriction operator of the sphere (but of course everything that follows extends trivially to arbitrary positively-curved compact hypersurfaces), that is

(defined initially on Schwartz functions), then

Stein-Tomas theorem: satisfies the a-priori inequality

for all exponents such that (and this is sharp, by the Knapp example).

There are a number of proofs of such statement; originally it was proven by Tomas for every exponent except the endpoint, and then Stein combined the proof of Tomas with his complex interpolation method to obtain the endpoint too (and this is still one of the finest examples of the power of the method around).
Carbery’s proof obtains the restricted endpoint inequality directly, and therefore obtains inequality (1) for all exponents < by interpolation of Lorentz spaces with the case (which is a trivial consequence of the Hausdorff-Young inequality).

In other words, Carbery proves that for any (Borel) measurable set one has

where the LHS is clearly the norm of the characteristic function . Notice that we could write the inequality equivalently as .

Bourgain's proof of the spherical maximal function theorem

Recently I have presented Stein’s proof of the boundedness of the spherical maximal function: it was in part III of a set of notes on basic Littlewood-Paley theory. Recall that the spherical maximal function is the operator

where denotes the spherical average at radius , that is

where denotes the spherical measure on the -dimensional sphere (we will omit the subscript from now on and just write since the dimension will not change throughout the arguments). We state Stein’s theorem for convenience:

Spherical maximal function theorem [Stein]: The maximal operator is bounded for any < .

There is however an alternative proof of the theorem due to Bourgain which is very nice and conceptually a bit simpler, in that instead of splitting the function into countably many dyadic frequency pieces it splits the spherical measure into two frequency pieces only. The other ingredients in the two proofs are otherwise pretty much the same: domination by the Hardy-Littlewood maximal function, Sobolev-type inequalities to control suprema by derivatives and oscillatory integral estimates for the Fourier transform of the spherical measure (and its derivative). However, Bourgain’s proof has an added bonus: remember that Stein’s argument essentially shows boundedness of the operator for every > quite directly; Bourgain’s argument, on the other hand, proves the restricted weak-type endpoint estimate for ! The latter means that for any measurable of finite (Lebesgue) measure we have

which is exactly the inequality but restricted to characteristic functions of sets (in the language of Lorentz spaces, it is the inequality). The downside of Bourgain’s argument is that it only works in dimension , and thus misses the dimension that is instead covered by Stein’s theorem.

It seems to me that, while Stein’s proof is well-known and has a number of presentations around, Bourgain’s proof is less well-known – it does not help that the original paper is impossible to find. As a consequence, I think it would be nice to share it here. This post is thus another tribute to Jean Bourgain, much in the same spirit as the posts (I – II) on his positional-notation trick for sets.

Oscillatory integrals II: several-variables phases

This is the second part of a two-part post on the theory of oscillatory integrals. In the first part we studied the theory of oscillatory integrals whose phases are functions of a single variable. In this post we will instead study the case in which the phase is a function of several variables (and we integrate in all of them). Here the theory becomes weaker because these objects can indeed have a worse behaviour. We will proceed by analogy following the same footsteps as in the single-variable case.
Part I

3. Oscillatory integrals in several variables

In the previous section we have analysed the situation for single variable phases, that is for integrals over (intervals of) . In this section, we will instead study the higher dimensional situation, when the phase is a function of several variables. Things are unfortunately generally not as nice as in the single variable case, as you will see.

In order to avoid having to worry about connected open sets of (see Exercise 18 for the sort of issues that arise in trying to deal with general open sets of ), in this section we will study mainly objects of the form

where has compact support. We have switched to for the phase to remind the reader of the fact that it is a function of several variables now.

3.1. Principle of non-stationary phase – several variables

The principle of non-stationary phase we saw in Section 2 of part I continues to hold in the several variables case.
Given a phase , we say that is a critical point of if

Proposition 8 (Principle of non-stationary phase – several variables) Let (that is, smooth and compactly supported) and let the phase be such that does not have critical points in the support of . Then for any we have

Oscillatory integrals I: single-variable phases

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.

0. Introduction

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.

1. Motivation

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

with , we have

This is the relationship we were looking for: it allows one to calculate the Fourier transform of directly from the radial information .

Marcinkiewicz-type multiplier theorem for q-variation (q > 1)

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:

Theorem 1 [Marcinkiewicz multiplier theorem]: Let be a function on such that

1. 
2. 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:

Theorem 2 [Coifman-Rubio de Francia-Semmes, ’88]: Let and let be a function on such that

1. 
2. 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).

Representing points in a set in positional-notation fashion (a trick by Bourgain): part II

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 .
Then

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 .

Representing points in a set in positional-notation fashion (a trick by Bourgain): part I

If you are reading this blog, you have probably heard that Jean Bourgain – one of the greatest analysts of the last century – has unfortunately passed away last December. It is fair to say that the progress of analysis will slow down significantly without him. I am not in any position to give a eulogy to this giant, but I thought it would be nice to commemorate him by talking occasionally on this blog about some of his many profound papers and his crazily inventive tricks. That’s something everybody agrees on: Bourgain was able to come up with a variety of insane tricks in a way that no one else is. The man was a problem solver and an overall magician: the first time you see one of his tricks, you don’t believe what’s happening in front of you. And that’s just the tricks part!

In this two-parts post I am going to talk about a certain trick that loosely speaking, involves representing points on an arbitrary set in a fashion similar to how integers are represented, say, in binary basis. I don’t know if this trick came straight out of Bourgain’s magical top hat or if he learned it from somewhere else; I haven’t seen it used elsewhere except for papers that cite Bourgain himself, so I’m inclined to attribute it to him – but please, correct me if I’m wrong.
Today we introduce the context for the trick (a famous lemma by Bourgain for maximal frequency projections on the real line) and present a toy version of the idea in a proof of the Rademacher-Menshov theorem. In the second part we will finally see the trick.

1. Ergodic averages along arithmetic sequences
First, some context. The trick I am going to talk about can be found in one of Bourgain’s major papers, that were among the ones cited in the motivation for his Fields medal prize. I am talking about the paper on a.e. convergence of ergodic averages along arithmetic sequences. The main result of that paper is stated as follows: let be an ergodic system, that is

1. is a probability on ;
2. satisfies for all -measurable sets (this is the invariance condition);
3. implies (this is the ergodicity condition).

Then the result is

Theorem: [Bourgain, ’89] Let be an ergodic system and let be a polynomial with integer coefficients. If with > 1, then the averages converge -a.e. as ; moreover, if is weakly mixing¹, we have more precisely

for -a.e. .

For comparison, the more classical pointwise ergodic theorem of Birkhoff states the same for the case and (notice this is the largest of the spaces because is finite), in which case the theorem is deduced as a consequence of the boundedness of the Hardy-Littlewood maximal function. The dense class to appeal to is roughly speaking , thanks to the ergodic theorem of Von Neumann, which states converges in norm for . However, the details are non-trivial. Heuristically, these ergodic theorems incarnate a quantitative version of the idea that the orbits fill up the entire space uniformly. I don’t want to enter into details because here I am just providing some context for those interested; there are plenty of introductions to ergodic theory where these results are covered in depth.

Kovač’s solution of the maximal Fourier restriction problem

About 2 years ago, Müller Ricci and Wright published a paper that opened a new line of investigation in the field of Fourier restriction: that is, the study of the pointwise meaning of the Fourier restriction operators. Here is a recount of a recent contribution to this problem that largely sorts it out.

1. Maximal Fourier Restriction
Recall that, given a smooth submanifold of with surface measure , the restriction operator is defined (initially) for Schwartz functions as

it is only after having proven an a-priori estimate such as that we can extend to an operator over the whole of , by density of the Schwartz functions. However, it is no longer clear what is the relationship between this new operator that has been operator-theoretically extended and the original operator that had a clear pointwise definition. In particular, a non-trivial question to ask is whether for -a.e. point we have

where is the ball of radius and center . Observe that the Lebesgue differentiation theorem already tells us that for a.e. element of in the Lebesgue sense the above holds; but the submanifold has Lebesgue measure zero, and therefore the differentiation theorem cannot give us any information. In this sense, the question above is about the structure of the set of the Lebesgue points of and can be reformulated as:

Q: can the complement of the set of Lebesgue points of contain a copy of the manifold ?

Basic Littlewood-Paley theory III: applications

This is the last part of a 3 part series on the basics of Littlewood-Paley theory. Today we discuss a couple of applications, that is Marcinkiewicz multiplier theorem and the boundedness of the spherical maximal function (the latter being an application of frequency decompositions in general, and not so much of square functions – though one appears, but only for estimates where one does not need the sophistication of Littlewood-Paley theory).
Part I: frequency projections
Part II: square functions

7. Applications of Littlewood-Paley theory

In this section we will present two applications of the Littlewood-Paley theory developed so far. You can find further applications in the exercises (see particularly Exercise 22 and Exercise 23).

7.1. Marcinkiewicz multipliers

Given an function , one can define the operator given by

for all . The operator is called a multiplier and the function is called the symbol of the multiplier¹. Since , Plancherel’s theorem shows that is a linear operator bounded in ; its definition can then be extended to functions (which are dense in ). A natural question to ask is: for which values of in is the operator an bounded operator? When is bounded in a certain space, we say that it is an –multiplier.

The operator introduced in Section 1 of the first post in this series is an example of a multiplier, with symbol . It is the linear operator that satisfies the formal identity . We have seen that it cannot be a (euclidean) Calderón-Zygmund operator, and thus in particular it cannot be a Hörmander-Mikhlin multiplier. This can be seen more directly by the fact that any Hörmander-Mikhlin condition of the form is clearly incompatible with the rescaling invariance of the symbol , which satisfies for any . However, the derivatives of actually satisfy some other superficially similar conditions that are of interest to us. Indeed, letting for simplicity, we can see for example that . When we can therefore argue that , and similarly when ; this shows that for any with one has

This condition is comparable with the corresponding Hörmander-Mikhlin condition only when , and is vastly different otherwise, being of product type (also notice that the inequality above is compatible with the rescaling invariance of , as it should be).