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This set of notes discusses aspects of one of the oldest questions in Fourier analysis, namely the nature of convergence of Fourier series.

If is an absolutely integrable function, its Fourier coefficients are defined by the formula

If is smooth, then the Fourier coefficients are absolutely summable, and we have the Fourier inversion formula where the series here is uniformly convergent. In particular, if we define the partial summation operators then converges uniformly to when is smooth.What if is not smooth, but merely lies in an class for some ? The Fourier coefficients remain well-defined, as do the partial summation operators . The question of convergence in norm is relatively easy to settle:

Exercise 1

- (i) If and , show that converges in norm to . (
Hint:first use the boundedness of the Hilbert transform to show that is bounded in uniformly in .)- (ii) If or , show that there exists such that the sequence is unbounded in (so in particular it certainly does not converge in norm to . (
Hint:first show that is not bounded in uniformly in , then apply the uniform boundedness principle in the contrapositive.)

The question of pointwise almost everywhere convergence turned out to be a significantly harder problem:

Theorem 2 (Pointwise almost everywhere convergence)

Note from Hölder’s inequality that contains for all , so Carleson’s theorem covers the case of Hunt’s theorem. We remark that the precise threshold near between Kolmogorov-type divergence results and Carleson-Hunt pointwise convergence results, in the category of Orlicz spaces, is still an active area of research; see this paper of Lie for further discussion.

Carleson’s theorem in particular was a surprisingly difficult result, lying just out of reach of classical methods (as we shall see later, the result is much easier if we smooth either the function or the summation method by a tiny bit). Nowadays we realise that the reason for this is that Carleson’s theorem essentially contains a *frequency modulation symmetry* in addition to the more familiar translation symmetry and dilation symmetry. This basically rules out the possibility of attacking Carleson’s theorem with tools such as Calderón-Zygmund theory or Littlewood-Paley theory, which respect the latter two symmetries but not the former. Instead, tools from “time-frequency analysis” that essentially respect all three symmetries should be employed. We will illustrate this by giving a relatively short proof of Carleson’s theorem due to Lacey and Thiele. (There are other proofs of Carleson’s theorem, including Carleson’s original proof, its modification by Hunt, and a later time-frequency proof by Fefferman; see Remark 18 below.)

Suppose one has a measure space and a sequence of operators that are bounded on some space, with . Suppose that on some dense subclass of functions in (e.g. continuous compactly supported functions, if the space is reasonable), one already knows that converges pointwise almost everywhere to some limit , for another bounded operator (e.g. could be the identity operator). What additional ingredient does one need to pass to the limit and conclude that converges almost everywhere to for *all* in (and not just for in a dense subclass)?

One standard way to proceed here is to study the *maximal operator*

and aim to establish a *weak-type maximal inequality*

for all (or all in the dense subclass), and some constant , where is the weak norm

A standard approximation argument using (1) then shows that will now indeed converge to pointwise almost everywhere for all in , and not just in the dense subclass. See for instance these lecture notes of mine, in which this method is used to deduce the Lebesgue differentiation theorem from the Hardy-Littlewood maximal inequality. This is by now a very standard approach to establishing pointwise almost everywhere convergence theorems, but it is natural to ask whether it is strictly necessary. In particular, is it possible to have a pointwise convergence result without being able to obtain a weak-type maximal inequality of the form (1)?

In the case of *norm* convergence (in which one asks for to converge to in the norm, rather than in the pointwise almost everywhere sense), the answer is no, thanks to the uniform boundedness principle, which among other things shows that norm convergence is only possible if one has the uniform bound

for some and all ; and conversely, if one has the uniform bound, and one has already established norm convergence of to on a dense subclass of , (2) will extend that norm convergence to all of .

Returning to pointwise almost everywhere convergence, the answer in general is “yes”. Consider for instance the rank one operators

from to . It is clear that converges pointwise almost everywhere to zero as for any , and the operators are uniformly bounded on , but the maximal function does not obey (1). One can modify this example in a number of ways to defeat almost any reasonable conjecture that something like (1) should be necessary for pointwise almost everywhere convergence.

In spite of this, a remarkable observation of Stein, now known as *Stein’s maximal principle*, asserts that the maximal inequality *is* necessary to prove pointwise almost everywhere convergence, if one is working on a compact group and the operators are translation invariant, and if the exponent is at most :

Theorem 1 (Stein maximal principle)Let be a compact group, let be a homogeneous space of with a finite Haar measure , let , and let be a sequence of bounded linear operators commuting with translations, such that converges pointwise almost everywhere for each . Then (1) holds.

This is not quite the most general vesion of the principle; some additional variants and generalisations are given in the original paper of Stein. For instance, one can replace the discrete sequence of operators with a continuous sequence without much difficulty. As a typical application of this principle, we see that Carleson’s celebrated theorem that the partial Fourier series of an function converge almost everywhere is in fact equivalent to the estimate

And unsurprisingly, most of the proofs of this (difficult) theorem have proceeded by first establishing (3), and Stein’s maximal principle strongly suggests that this is the optimal way to try to prove this theorem.

On the other hand, the theorem does fail for , and almost everywhere convergence results in for can be proven by other methods than weak estimates. For instance, the convergence of Bochner-Riesz multipliers in for any (and for in the range predicted by the Bochner-Riesz conjecture) was verified for by Carbery, Rubio de Francia, and Vega, despite the fact that the weak of even a *single* Bochner-Riesz multiplier, let alone the maximal function, has still not been completely verified in this range. (Carbery, Rubio de Francia and Vega use weighted estimates for the maximal Bochner-Riesz operator, rather than type estimates.) For , though, Stein’s principle (after localising to a torus) does apply, though, and pointwise almost everywhere convergence of Bochner-Riesz means is equivalent to the weak estimate (1).

Stein’s principle is restricted to compact groups (such as the torus or the rotation group ) and their homogeneous spaces (such as the torus again, or the sphere ). As stated, the principle fails in the noncompact setting; for instance, in , the convolution operators are such that converges pointwise almost everywhere to zero for every , but the maximal function is not of weak-type . However, in many applications on non-compact domains, the are “localised” enough that one can transfer from a non-compact setting to a compact setting and then apply Stein’s principle. For instance, Carleson’s theorem on the real line is equivalent to Carleson’s theorem on the circle (due to the localisation of the Dirichlet kernels), which as discussed before is equivalent to the estimate (3) on the circle, which by a scaling argument is equivalent to the analogous estimate on the real line .

Stein’s argument from his 1961 paper can be viewed nowadays as an application of the probabilistic method; starting with a sequence of increasingly bad counterexamples to the maximal inequality (1), one randomly combines them together to create a single “infinitely bad” counterexample. To make this idea work, Stein employs two basic ideas:

- The
*random rotations (or random translations) trick*. Given a subset of of small but positive measure, one can randomly select about translates of that cover most of . - The
*random sums trick*Given a collection of signed functions that may possibly cancel each other in a deterministic sum , one can perform a random sum instead to obtain a random function whose magnitude will usually be comparable to the square function ; this can be made rigorous by concentration of measure results, such as Khintchine’s inequality.

These ideas have since been used repeatedly in harmonic analysis. For instance, I used the random rotations trick in a recent paper with Jordan Ellenberg and Richard Oberlin on Kakeya-type estimates in finite fields. The random sums trick is by now a standard tool to build various counterexamples to estimates (or to convergence results) in harmonic analysis, for instance being used by Fefferman in his famous paper disproving the boundedness of the ball multiplier on for , . Another use of the random sum trick is to show that Theorem 1 fails once ; see Stein’s original paper for details.

Another use of the random rotations trick, closely related to Theorem 1, is the *Nikishin-Stein factorisation theorem*. Here is Stein’s formulation of this theorem:

Theorem 2 (Stein factorisation theorem)Let be a compact group, let be a homogeneous space of with a finite Haar measure , let and , and let be a bounded linear operator commuting with translations and obeying the estimatefor all and some . Then also maps to , with

for all , with depending only on .

This result is trivial with , but becomes useful when . In this regime, the translation invariance allows one to freely “upgrade” a strong-type result to a weak-type result. In other words, bounded linear operators from to automatically factor through the inclusion , which helps explain the name “factorisation theorem”. Factorisation theory has been developed further by many authors, including Maurey and Pisier.

Stein’s factorisation theorem (or more precisely, a variant of it) is useful in the theory of Kakeya and restriction theorems in Euclidean space, as first observed by Bourgain.

In 1970, Nikishin obtained the following generalisation of Stein’s factorisation theorem in which the translation-invariance hypothesis can be dropped, at the cost of excluding a set of small measure:

Theorem 3 (Nikishin-Stein factorisation theorem)Let be a finite measure space, let and , and let be a bounded linear operator obeying the estimatefor all and some . Then for any , there exists a subset of of measure at most such that

One can recover Theorem 2 from Theorem 3 by an averaging argument to eliminate the exceptional set; we omit the details.

If one has a sequence of real numbers , it is unambiguous what it means for that sequence to converge to a limit : it means that for every , there exists an such that for all . Similarly for a sequence of complex numbers converging to a limit .

More generally, if one has a sequence of -dimensional vectors in a real vector space or complex vector space , it is also unambiguous what it means for that sequence to converge to a limit or ; it means that for every , there exists an such that for all . Here, the norm of a vector can be chosen to be the Euclidean norm , the supremum norm , or any other number of norms, but for the purposes of convergence, these norms are all *equivalent*; a sequence of vectors converges in the Euclidean norm if and only if it converges in the supremum norm, and similarly for any other two norms on the finite-dimensional space or .

If however one has a sequence of functions or on a common domain , and a putative limit or , there can now be many different ways in which the sequence may or may not converge to the limit . (One could also consider convergence of functions on different domains , but we will not discuss this issue at all here.) This is contrast with the situation with scalars or (which corresponds to the case when is a single point) or vectors (which corresponds to the case when is a finite set such as ). Once becomes infinite, the functions acquire an infinite number of degrees of freedom, and this allows them to approach in any number of inequivalent ways.

What different types of convergence are there? As an undergraduate, one learns of the following two basic modes of convergence:

- We say that converges to pointwise if, for every , converges to . In other words, for every and , there exists (that depends on
*both*and ) such that whenever . - We say that converges to uniformly if, for every , there exists such that for every , for every . The difference between uniform convergence and pointwise convergence is that with the former, the time at which must be permanently -close to is not permitted to depend on , but must instead be chosen uniformly in .

Uniform convergence implies pointwise convergence, but not conversely. A typical example: the functions defined by converge pointwise to the zero function , but not uniformly.

However, pointwise and uniform convergence are only two of dozens of many other modes of convergence that are of importance in analysis. We will not attempt to exhaustively enumerate these modes here (but see this Wikipedia page, and see also these 245B notes on strong and weak convergence). We will, however, discuss some of the modes of convergence that arise from measure theory, when the domain is equipped with the structure of a measure space , and the functions (and their limit ) are measurable with respect to this space. In this context, we have some additional modes of convergence:

- We say that converges to pointwise almost everywhere if, for (-)almost everywhere , converges to .
- We say that converges to
*uniformly almost everywhere*,*essentially uniformly*, or*in norm*if, for every , there exists such that for every , for -almost every . - We say that converges to
*almost uniformly*if, for every , there exists an exceptional set of measure such that converges uniformly to on the complement of . - We say that converges to in norm if the quantity converges to as .
- We say that converges to in measure if, for every , the measures converge to zero as .

Observe that each of these five modes of convergence is unaffected if one modifies or on a set of measure zero. In contrast, the pointwise and uniform modes of convergence can be affected if one modifies or even on a single point.

Remark 1In the context of probability theory, in which and are interpreted as random variables, convergence in norm is often referred to as convergence in mean, pointwise convergence almost everywhere is often referred to as almost sure convergence, and convergence in measure is often referred to as convergence in probability.

Exercise 1 (Linearity of convergence)Let be a measure space, let be sequences of measurable functions, and let be measurable functions.

- Show that converges to along one of the above seven modes of convergence if and only if converges to along the same mode.
- If converges to along one of the above seven modes of convergence, and converges to along the same mode, show that converges to along the same mode, and that converges to along the same mode for any .
- (Squeeze test) If converges to along one of the above seven modes, and pointwise for each , show that converges to along the same mode.

We have some easy implications between modes:

Exercise 2 (Easy implications)Let be a measure space, and let and be measurable functions.

- If converges to uniformly, then converges to pointwise.
- If converges to uniformly, then converges to in norm. Conversely, if converges to in norm, then converges to uniformly outside of a null set (i.e. there exists a null set such that the restriction of to the complement of converges to the restriction of ).
- If converges to in norm, then converges to almost uniformly.
- If converges to almost uniformly, then converges to pointwise almost everywhere.
- If converges to pointwise, then converges to pointwise almost everywhere.
- If converges to in norm, then converges to in measure.
- If converges to almost uniformly, then converges to in measure.

The reader is encouraged to draw a diagram that summarises the logical implications between the seven modes of convergence that the above exercise describes.

We give four key examples that distinguish between these modes, in the case when is the real line with Lebesgue measure. The first three of these examples already were introduced in the previous set of notes.

Example 1 (Escape to horizontal infinity)Let . Then converges to zero pointwise (and thus, pointwise almost everywhere), but not uniformly, in norm, almost uniformly, in norm, or in measure.

Example 2 (Escape to width infinity)Let . Then converges to zero uniformly (and thus, pointwise, pointwise almost everywhere, in norm, almost uniformly, and in measure), but not in norm.

Example 3 (Escape to vertical infinity)Let . Then converges to zero pointwise (and thus, pointwise almost everywhere) and almost uniformly (and hence in measure), but not uniformly, in norm, or in norm.

Example 4 (Typewriter sequence)Let be defined by the formulawhenever and . This is a sequence of indicator functions of intervals of decreasing length, marching across the unit interval over and over again. Then converges to zero in measure and in norm, but not pointwise almost everywhere (and hence also not pointwise, not almost uniformly, nor in norm, nor uniformly).

Remark 2Thenormof a measurable function is defined to the infimum of all the quantities that areessential upper boundsfor in the sense that for almost every . Then converges to in norm if and only if as . The and norms are part of the larger family of norms, which we will study in more detail in 245B.

One particular advantage of convergence is that, in the case when the are absolutely integrable, it implies convergence of the integrals,

as one sees from the triangle inequality. Unfortunately, none of the other modes of convergence automatically imply this convergence of the integral, as the above examples show.

The purpose of these notes is to compare these modes of convergence with each other. Unfortunately, the relationship between these modes is not particularly simple; unlike the situation with pointwise and uniform convergence, one cannot simply rank these modes in a linear order from strongest to weakest. This is ultimately because the different modes react in different ways to the three “escape to infinity” scenarios described above, as well as to the “typewriter” behaviour when a single set is “overwritten” many times. On the other hand, if one imposes some additional assumptions to shut down one or more of these escape to infinity scenarios, such as a finite measure hypothesis or a uniform integrability hypothesis, then one can obtain some additional implications between the different modes.

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