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)}} .

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