## 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 $(X,T,\mu)$ be an ergodic system, that is

1. $\mu$ is a probability on $X$;
2. $T: X \to X$ satisfies $\mu(T^{-1} A) = \mu(A)$ for all $\mu$-measurable sets $A$ (this is the invariance condition);
3. $T^{-1} A = A$ implies $\mu(A) = 0 \text{ or } 1$ (this is the ergodicity condition).

Then the result is

Theorem: [Bourgain, ’89] Let $(X,T,\mu)$ be an ergodic system and let $p(n)$ be a polynomial with integer coefficients. If $f \in L^q(d\mu)$ with ${q}$ > 1, then the averages $A_N f(x) := \frac{1}{N}\sum_{n=1}^{N}f(T^{p(n)} x)$ converge $\mu$-a.e. as $N \to \infty$; moreover, if ${T}$ is weakly mixing1, we have more precisely

$\displaystyle \lim_{N \to \infty} A_N f(x) = \int_X f d\mu$

for $\mu$-a.e. ${x}$.

For comparison, the more classical pointwise ergodic theorem of Birkhoff states the same for the case $p(n) = n$ and $f \in L^1(d\mu)$ (notice this is the largest of the $L^p(X,d\mu)$ spaces because $\mu$ is finite), in which case the theorem is deduced as a consequence of the $L^1 \to L^{1,\infty}$ boundedness of the Hardy-Littlewood maximal function. The dense class to appeal to is roughly speaking $L^2(X,d\mu)$, thanks to the ergodic theorem of Von Neumann, which states $A_N f$ converges in $L^2$ norm for $f \in L^2(X,d\mu)$. However, the details are non-trivial. Heuristically, these ergodic theorems incarnate a quantitative version of the idea that the orbits $\{T^n x\}_{n\in\mathbb{N}}$ fill up the entire space ${X}$ 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.

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## 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 $\Sigma$ of $\mathbb{R}^d$ with surface measure $d\sigma$, the restriction operator ${R}$ is defined (initially) for Schwartz functions as

$\displaystyle f \mapsto Rf:= \widehat{f}\Big|_{\Sigma};$

it is only after having proven an a-priori estimate such as $\|Rf\|_{L^q(\Sigma,d\sigma)} \lesssim \|f\|_{L^p(\mathbb{R}^d)}$ that we can extend ${R}$ to an operator over the whole of $L^p(\mathbb{R}^d)$, 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 $d\sigma$-a.e. point $\xi \in \Sigma$ we have

$\displaystyle \lim_{r \to 0} \frac{1}{|B(0,r)|} \int_{\eta \in B(0,r)} |\widehat{f}(\xi - \eta)| d\eta = \widehat{f}(\xi), \ \ \ \ \ (1)$

where $B(0,r)$ is the ball of radius ${r}$ and center ${0}$. Observe that the Lebesgue differentiation theorem already tells us that for a.e. element of $\mathbb{R}^d$ in the Lebesgue sense the above holds; but the submanifold $\Sigma$ 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 $\widehat{f}$ and can be reformulated as:

Q: can the complement of the set of Lebesgue points of $\widehat{f}$ contain a copy of the manifold $\Sigma$?

## 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 $L^2$ 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 ${L^\infty (\mathbb{R}^d)}$ function ${m}$, one can define the operator ${T_m}$ given by

$\displaystyle \widehat{T_m f}(\xi) := m(\xi) \widehat{f}(\xi)$

for all ${f \in L^2(\mathbb{R}^d)}$. The operator ${T_m}$ is called a multiplier and the function ${m}$ is called the symbol of the multiplier1. Since ${m \in L^\infty}$, Plancherel’s theorem shows that ${T_m}$ is a linear operator bounded in ${L^2}$; its definition can then be extended to ${L^2 \cap L^p}$ functions (which are dense in ${L^p}$). A natural question to ask is: for which values of ${p}$ in ${1 \leq p \leq \infty}$ is the operator ${T_m}$ an ${L^p \rightarrow L^p}$ bounded operator? When ${T_m}$ is bounded in a certain ${L^p}$ space, we say that it is an ${L^p}$multiplier.

The operator ${T_m}$ introduced in Section 1 of the first post in this series is an example of a multiplier, with symbol ${m(\xi,\tau) = \tau / (\tau - 2\pi i |\xi|^2)}$. It is the linear operator that satisfies the formal identity $T \circ (\partial_t - \Delta) = \partial_t$. 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 ${|\partial^{\alpha}m(\xi,\tau)| \lesssim_\alpha |(\xi,\tau)|^{-|\alpha|} = (|\xi|^2 + \tau^2)^{-|\alpha|/2}}$ is clearly incompatible with the rescaling invariance of the symbol ${m}$, which satisfies ${m(\lambda \xi, \lambda^2 \tau) = m(\xi,\tau)}$ for any ${\lambda \neq 0}$. However, the derivatives of ${m}$ actually satisfy some other superficially similar conditions that are of interest to us. Indeed, letting ${(\xi,\tau) \in \mathbb{R}^2}$ for simplicity, we can see for example that ${\partial_\xi \partial_\tau m(\xi, \tau) = \lambda^3 \partial_\xi \partial_\tau m(\lambda\xi, \lambda^2\tau)}$. When ${|\tau|\lesssim |\xi|^2}$ we can therefore argue that ${|\partial_\xi \partial_\tau m(\xi, \tau)| = |\xi|^{-3} |\partial_\xi \partial_\tau m(1, \tau |\xi|^{-2})| \lesssim |\xi|^{-1} |\tau|^{-1} \sup_{|\eta|\lesssim 1} |\partial_\xi \partial_\tau m(1, \eta)|}$, and similarly when ${|\tau|\gtrsim |\xi|^2}$; this shows that for any ${(\xi, \tau)}$ with ${\xi,\tau \neq 0}$ one has

$\displaystyle |\partial_\xi \partial_\tau m(\xi, \tau)| \lesssim |\xi|^{-1} |\tau|^{-1}.$

This condition is comparable with the corresponding Hörmander-Mikhlin condition only when ${|\xi| \sim |\tau|}$, and is vastly different otherwise, being of product type (also notice that the inequality above is compatible with the rescaling invariance of ${m}$, as it should be).
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## Basic Littlewood-Paley theory II: square functions

This is the second part of the series on basic Littlewood-Paley theory, which has been extracted from some lecture notes I wrote for a masterclass. In this part we will prove the Littlewood-Paley inequalities, namely that for any ${1 < p < \infty}$ it holds that

$\displaystyle \|f\|_{L^p (\mathbb{R})} \sim_p \Big\|\Big(\sum_{j \in \mathbb{Z}} |\Delta_j f|^2 \Big)^{1/2}\Big\|_{L^p (\mathbb{R})}. \ \ \ \ \ (\dagger)$

This time there are also plenty more exercises, some of which I think are fairly interesting (one of them is a theorem of Rudin in disguise).
Part I: frequency projections.

4. Smooth square function

In this subsection we will consider a variant of the square function appearing at the right-hand side of ($\dagger$) where we replace the frequency projections ${\Delta_j}$ by better behaved ones.

Let ${\psi}$ denote a smooth function with the properties that ${\psi}$ is compactly supported in the intervals ${[-4,-1/2] \cup [1/2, 4]}$ and is identically equal to ${1}$ on the intervals ${[-2,-1] \cup [1,2]}$. We define the smooth frequency projections ${\widetilde{\Delta}_j}$ by stipulating

$\displaystyle \widehat{\widetilde{\Delta}_j f}(\xi) := \psi(2^{-j} \xi) \widehat{f}(\xi);$

notice that the function ${\psi(2^{-j} \xi)}$ is supported in ${[-2^{j+2},-2^{j-1}] \cup [2^{j-1}, 2^{j+2}]}$ and identically ${1}$ in ${[-2^{j+1},-2^{j}] \cup [2^{j}, 2^{j+1}]}$. The reason why such projections are better behaved resides in the fact that the functions ${\psi(2^{-j}\xi)}$ are now smooth, unlike the characteristic functions ${\mathbf{1}_{[2^j,2^{j+1}]}}$. Indeed, they are actually Schwartz functions and you can see by Fourier inversion formula that ${\widetilde{\Delta}_j f = f \ast (2^{j} \widehat{\psi}(2^{j}\cdot))}$; the convolution kernel ${2^{j} \widehat{\psi}(2^{j}\cdot)}$ is uniformly in ${L^1}$ and therefore the operator is trivially ${L^p \rightarrow L^p}$ bounded for any ${1 \leq p \leq \infty}$ by Young’s inequality, without having to resort to the boundedness of the Hilbert transform.
We will show that the following smooth analogue of (one half of) ($\dagger$) is true (you can study the other half in Exercise 6).

Proposition 3 Let ${\widetilde{S}}$ denote the square function

$\displaystyle \widetilde{S}f := \Big(\sum_{j \in \mathbb{Z}} \big|\widetilde{\Delta}_j f \big|^2\Big)^{1/2}.$

Then for any ${1 < p < \infty}$ we have that the inequality

$\displaystyle \big\|\widetilde{S}f\big\|_{L^p(\mathbb{R})} \lesssim_p \|f\|_{L^p(\mathbb{R})} \ \ \ \ \ (1)$

holds for any ${f \in L^p(\mathbb{R})}$.

We will give two proofs of this fact, to illustrate different techniques. We remark that the boundedness will depend on the smoothness and the support properties of ${\psi}$ only, and as such extends to a larger class of square functions.
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## Quadratically modulated Bilinear Hilbert transform

Here is a simple but surprising fact.

Recall that the Hilbert transform $Hf(x) := p.v. \int f(x-t) \frac{dt}{t}$ is $L^p \to L^p$ bounded for all ${1 < p < \infty}$ (and even $L^1 \to L^{1,\infty}$ bounded, of course). The quadratically modulated Hilbert transform is the operator

$\displaystyle H_q f(x) := p.v. \int \int f(x-t) e^{-i t^2} \frac{dt}{t};$

this operator is also known to be $L^p \to L^p$ bounded for all ${1 < p < \infty}$, but the proof is no corollary of that for $H$, it's a different beast requiring oscillatory integral techniques and almost orthogonality and is due to Ricci and Stein (interestingly though, $H_q$ is also $L^1 \to L^{1,\infty}$ bounded, and this can indeed be obtained by a clever adaptation of Calderón-Zygmund theory due to Chanillo and Christ).

The bilinear Hilbert transform instead is the operator

$\displaystyle BHT(f,g)(x) := p.v. \int \int f(x-t)g(x+t)\frac{dt}{t}$

and it is known, thanks to foundational work of Lacey and Thiele, to be $L^p \times L^q \to L^r$ bounded at least in the range given by $p,q>1, r>2/3$ with the exponents satisfying the Hölder condition $1/p + 1/q = 1/r$ (this condition is strictly necessary to have boundedness, due to the scaling invariance of the operator). This operator has an interesting modulation invariance (corresponding to the fact that its bilinear multiplier is $\mathrm{sgn}(\xi - \eta)$, which is invariant with respect to translations along the diagonal): indeed, if $\mathrm{Mod}_{\theta}$ denotes the modulation operator $\mathrm{Mod}_{\theta} f(x) := e^{- i \theta x} f(x)$, we have

$\displaystyle BHT(\mathrm{Mod}_{\theta}f,\mathrm{Mod}_{\theta}g) = \mathrm{Mod}_{2 \theta} BHT(f,g);$

it is this fact that suggests one should use time-frequency analysis to deal with this operator.
Now, analogously to the linear case, one can consider the quadratically modulated bilinear Hilbert transform, given simply by

$\displaystyle BHT_q(f,g)(x) := p.v. \int \int f(x-t)g(x+t) e^{-i t^2} \frac{dt}{t}.$

One might be tempted to think, by analogy, that this operator is harder to bound than $BHT$ – at least, I would naively think so at first sight. However, due to the particular structure of the bilinear Hilbert transform, the boundedness of $BHT_q$ is a trivial corollary of that of $BHT$! Indeed, this is due to the trivial polynomial identity

$\displaystyle (x+t)^2 + (x-t)^2 = 2x^2 + 2t^2;$

thus if $\mathrm{QMod}_{\theta}$ denotes the quadratic modulation operator $\mathrm{QMod}_{\theta}f(x) = e^{-i \theta x^2} f(x)$ we have

\displaystyle \begin{aligned} BHT_q(f,g)(x) = & \int f(x-t)g(x+t) e^{-it^2} \frac{dt}{t} \\ = & \int f(x-t)g(x+t) e^{ix^2}e^{-i(x+t)^2/2}e^{-i(x-t)^2/2} \frac{dt}{t} \\ = & e^{ix^2}\int e^{-i(x-t)^2/2}f(x-t)e^{-i(x+t)^2/2}g(x+t) \frac{dt}{t} \\ = & \big[ \mathrm{QMod}_{-1} BHT( \mathrm{QMod}_{1/2} f, \mathrm{QMod}_{1/2} g )\big](x). \end{aligned}

Of course this trick is limited to quadratic modulations, so for example already the cubic modulation of $BHT$

is non-trivial to bound (but the boundedness of the cubic modulation of the trilinear Hilbert transform would again be a trivial consequence of the boundedness of the trilinear Hilbert transform itself… too bad we don’t know if it is bounded at all!). Polynomial modulations of bilinear singular integrals (thus a bilinear analogue of the Ricci-Stein work) have been shown to be bounded by Christ, Li, Tao and Thiele in “On multilinear oscillatory integrals, nonsingular and singular“.

UPDATE: Interesting synchronicity just happened: today Dong, Maldague and Villano have uploaded on ArXiv their paper “Special cases of power decay in multilinear oscillatory integrals” in which they extend the work of Christ, Li, Tao and Thiele to some special cases that were left out. Maybe I should check my email for the arXiv digest before posting next time.

## Basic Littlewood-Paley theory I: frequency projections

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 ${f}$ 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 ${\mathbb{R}}$ given by

$\displaystyle f = \sum_{j \in \mathbb{Z}} \Delta_j f,$

where ${\Delta_j f}$ denotes the operator

$\displaystyle \Delta_j f (x) := \int_{\{\xi \in \mathbb{R} : 2^j \leq |\xi| < 2^{j+1}\}} \widehat{f}(\xi) e^{2\pi i \xi \cdot x} d\xi,$

commonly referred to as a (dyadic) frequency projection. Thus ${\Delta_j f}$ represents the portion of ${f}$ with frequencies of magnitude ${\sim 2^j}$. The Fourier inversion formula can be used to justify the above decomposition if, for example, ${f \in L^2(\mathbb{R})}$. Heuristically, since any two ${\Delta_j f, \Delta_{k} f}$ oscillate at significantly different frequencies when ${|j-k|}$ is large, we would expect that for most ${x}$‘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 ${|f|}$ should behave “most of the time” like ${\Big(\sum_{j \in \mathbb{Z}} |\Delta_j f|^2 \Big)^{1/2}}$ (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 ${L^p}$-norms: more precisely, we will show that for any ${1 < p < \infty}$ it holds that

$\displaystyle \|f\|_{L^p (\mathbb{R})} \sim_p \Big\|\Big(\sum_{j \in \mathbb{Z}} |\Delta_j f|^2 \Big)^{1/2}\Big\|_{L^p (\mathbb{R})}. \ \ \ \ \ (\dagger)$

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 ${p=2}$ 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 ${L^p}$ spaces when ${p\neq 2}$.

In developing a framework that allows to prove ($\dagger$) 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 ${\mathscr{M}_{\mathbb{S}^{d-1}}}$ (almost verbatim the one on Tao’s blog).

Notation: We will use ${A \lesssim B}$ to denote the estimate ${A \leq C B}$ where ${C>0}$ is some absolute constant, and ${A\sim B}$ to denote the fact that ${A \lesssim B \lesssim A}$. If the constant ${C}$ depends on a list of parameters ${L}$ we will write ${A \lesssim_L B}$.

• #### marcov 14:35 on 08/03/2019 Permalink | Reply

Dyatlov uploaded “An introduction to the Fractal Uncertainty Principle” to the arXiv today. Looking forward to reading this

## A cute combinatorial result of Santaló

There is a nice result due to Santaló that says that if a (finite) collection of axis-parallel rectangles is such that any small subcollection is aligned, then the whole collection is aligned. This is kind of surprising at first, because the condition only says that there is a line, but this line might be different for any choice of subcollection. The precise statement is as follows:

Theorem. Let $\mathcal{R}$ be a collection of rectangles with sides parallel to the axes (possibly intersecting). If for every choice of 6 rectangles of $\mathcal{R}$ there exists a line intersecting all $6$ of them, then there exists a line intersecting all rectangles of $\mathcal{R}$ at once.

To be precise, I should clarify that by line intersection it is meant intersection with the interior of the rectangle – so a line touching only the boundary is not allowed. The number 6 doesn’t have any special esoteric meaning here, to the best of my understanding – it just makes the argument work.
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## Recommended reads #1

This is a very short digression: I found Prof. Matt Strassler’s blog very intriguing. Here are a couple of links:

It attemps (and imho succeds at ) an introduction to the basic concepts of the Standard Model and the mechanism by which particles acquire mass through the Higgs field. It’s presented in a very intuitive way, while preserving enough of the mathematical structure. All the math is actually pretty elementary and there are some white lies here and there, which I think is a good thing. When you’re introducing someone to a subject, you want to sweep the dust under the carpet at first.
All in all, this much quality in science popularization is very rare I think.

## Christ’s result on near-equality in Riesz-Sobolev inequality

Pdf: link.

It’s finally time to address one of Christ’s papers I talked about in the previous two blogposts. As mentioned there, I’ve chosen to read the one about the near-equality in the Riesz-Sobolev inequality because it seems the more approachable, while still containing one very interesting idea: exploiting the additive structure lurking behind the inequality via Freiman’s theorem.

1. Elaborate an attack strategy

Everything is in dimension ${d=1}$ and some details of the proof are specific to this dimension and don’t extend to higher dimensions. I’ll stick to Christ’s notation.

Recall that the Riesz-Sobolev inequality is

$\displaystyle \boxed{\left\langle \chi_{A} \ast \chi_{B}, \chi_{C}\right\rangle \leq \left\langle \chi_{A^\ast} \ast \chi_{B^\ast}, \chi_{C^\ast}\right\rangle} \ \ \ \ \ (1)$

and its extremizers – which exist under the hypothesis that the sizes are all comparable – are intervals, i.e. the intervals are the only sets that realize equality in (1). See previous post for further details. The aim of paper [ChRS] is to prove that whenever ${\left\langle \chi_{A} \ast \chi_{B}, \chi_{C}\right\rangle}$ is suitably close to ${\left\langle \chi_{A^\ast} \ast \chi_{B^\ast}, \chi_{C^\ast}\right\rangle}$ (i.e. we nearly have equality) then the sets ${A,B,C}$ are nearly intervals themselves.

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