# 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

$\displaystyle \mathscr{M}_{\mathbb{S}^{d-1}} f(x) := \sup_{t > 0} |A_t f(x)|,$

where $A_t$ denotes the spherical average at radius ${t}$, that is

$\displaystyle A_t f(x) := \int_{\mathbb{S}^{d-1}} f(x - t\omega) d\sigma_{d-1}(\omega),$

where $d\sigma_{d-1}$ denotes the spherical measure on the $(d-1)$-dimensional sphere (we will omit the subscript from now on and just write $d\sigma$ since the dimension will not change throughout the arguments). We state Stein’s theorem for convenience:

Spherical maximal function theorem [Stein]: The maximal operator $\mathcal{M}_{\mathbb{S}^{d-1}}$ is $L^p(\mathbb{R}^d) \to L^p(\mathbb{R}^d)$ bounded for any $\frac{d}{d-1}$ < $p \leq \infty$.

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 $L^p \to L^p$ boundedness of the operator for every $2 \geq p$ > $\frac{d}{d-1}$ quite directly; Bourgain’s argument, on the other hand, proves the restricted weak-type endpoint estimate for $\mathcal{M}_{\mathbb{S}^{d-1}}$! The latter means that for any measurable $E$ of finite (Lebesgue) measure we have

$\displaystyle |\{x \in \mathbb{R}^d \; : \; \mathcal{M}_{\mathbb{S}^{d-1}}\mathbf{1}_E(x) > \alpha \}| \lesssim \frac{|E|}{\alpha^{d/(d-1)}}, \ \ \ \ \ \ (1)$

which is exactly the $L^{d/(d-1)} \to L^{d/(d-1),\infty}$ inequality but restricted to characteristic functions of sets (in the language of Lorentz spaces, it is the $L^{d/(d-1),1} \to L^{d/(d-1),\infty}$ inequality). The downside of Bourgain’s argument is that it only works in dimension $d \geq 4$, and thus misses the dimension $d=3$ 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 (III) on his positional-notation trick for sets.

# 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 an account 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 the relationship is 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).