# 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 $L^p$ 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 $R$ denotes the Fourier restriction operator of the sphere $\mathbb{S}^{d-1}$ (but of course everything that follows extends trivially to arbitrary positively-curved compact hypersurfaces), that is

$\displaystyle Rf = \widehat{f} \,\big|_{\mathbb{S}^{d-1}}$

(defined initially on Schwartz functions), then

Stein-Tomas theorem: $R$ satisfies the a-priori inequality

$\displaystyle \|Rf\|_{L^2(\mathbb{S}^{d-1},d\sigma)} \lesssim_p \|f\|_{L^p(\mathbb{R}^d)} \ \ \ \ \ \ (1)$

for all exponents ${p}$ such that $1 \leq p \leq \frac{2(d+1)}{d+3}$ (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 $1 \leq p$ < $\frac{2(d+1)}{d+3}$ by interpolation of Lorentz spaces with the $p=1$ case (which is a trivial consequence of the Hausdorff-Young inequality).

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

$\displaystyle \|R \mathbf{1}_{E}\|_{L^2(\mathbb{S}^{d-1},d\sigma)} \lesssim |E|^{\frac{d+3}{2(d+1)}}, \ \ \ \ \ \ (2)$

where the LHS is clearly the $L^{2(d+1)/(d+3)}$ norm of the characteristic function $\mathbf{1}_E$. Notice that we could write the inequality equivalently as $\|\widehat{\mathbf{1}_{E}}\|_{L^2(\mathbb{S}^{d-1},d\sigma)} \lesssim |E|^{\frac{d+3}{2(d+1)}}$.

# 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.

# Lorentz spaces basics & interpolation

(Updated with endpoint ${q = \infty}$)

I’ve written down an almost self contained exposition of basic properties of Lorentz spaces. I’ve found the sources on the subject to leave something to be desired, and I grew a bit confused at the beginning. Therefore this relatively short note (I might be ruining someone’s assignments out there, but I think the pros of writing down everything in one place balance the cons).

1. Lorentz spaces

In the following take ${1< p,q < \infty}$ otherwise specified, and ${(X, |\cdot|)}$ a ${\sigma}$-finite measure space with no atoms.

The usual definition of Lorentz space is as follows:

Definition 1 The space ${L^{p,q}(X)}$ is the space of measurable functions ${f}$ such that

$\displaystyle \|f\|_{L^{p,q}(X)}:= \left(\int_{0}^{\infty}{t^{q/p}{ f^\ast (t)}^q}\,\frac{dt}{t}\right)^{1/q} < \infty,$

where ${f^\ast}$ is the decreasing rearrangement [I] of ${f}$. If ${q=\infty}$ then define instead

$\displaystyle \|f\|_{L^{p,\infty}(X)} := \sup_{t}{t^{1/p} f^\ast (t)} < \infty.$