# 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.