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A Remark on reverse Littlewood-Paley, restriction and Kakeya

We show that a certain conjectured optimal reverse Littlewood- Paley inequality would, if true, imply sharp results for the Kakeya maximal function, the Bochner-Riesz means and the Fourier restriction operator.


Introduction
Let δ > 0 be a small parameter and let Φ : R → R be a smooth function of compact support in [−1, 1] satisfying |Φ (k) (t)| ≤ C k for all k ∈ N. Define the Fourier multiplier operator S δ on R n by We decompose the δ-neighbourhood {ξ : |ξ| − 1 ≤ δ} of the unit sphere S n−1 into coin-shaped pieces E α of tangential dimensions δ 1/2 × · · · × δ 1/2 and radial dimension δ, and correspondingly the operator S δ = α S α where S α is a Fourier multiplier operator with smooth multiplier φ α supported in and adapted to E α . We consider the reverse Littlewood-Paley inequality where C n is supposed to be independent of δ and f . When n = 2 (and thus 2n/(n − 1) = 4), Fefferman in [11] proved that this inequality is true since we can multiply out the L 4 norm and observe that the algebraic differences E α − E α ′ are essentially disjoint as α = α ′ vary. In higher dimensions it remains open.
More generally one might ask whether for q ≥ 2 we have where for 2 ≤ q ≤ 2n/(n − 1) we take r = 2 and when q ≥ 2n/(n − 1) we take r ′ = q(n − 1)/n (so that when q = ∞ we have r = 1). The issue of such inequalities with the order the mixed norms reversed, first proposed by Bonami and Garrigós, has recently been studied in [3] where they are termed l r -decoupling inequalities.
There is a maximal function relevant to the study of S δ , the so-called Nikodym maximal function, for which Fefferman also proved in [11] optimal L 2 bounds in two dimensions (in slightly disguised form). By what are very familiar arguments (see for example [8], [9]) these two ingredients can be combined to prove the optimal two-dimensional result for the operators S δ .
In this note we show that (a variant of) (1) actually implies the correct L n bound for the maximal function, and thus (1) together with its variant gives the optimal Bochner-Riesz multiplier result in all dimensions upon combining (1) with the maximal function estimate in the familiar way.
For ease of exposition we choose to work in the alternative and essentially equivalent setting of the extension problem for the Fourier transform. Since the kernels of S δ and S α in (1) are essentially localised at scale 1/δ, (1) is equivalent to the corresponding local inequality and thus to 1 where g is a smooth function defined on S n−1 , σ is the Lebesgue measure on S n−1 , g α = gχ Eα , the E α are spherical caps of radius δ 1/2 decomposing S n−1 and where is the Fourier transform of the density hdσ. Note that the inverse Fourier transform of gdσ Φ(δ·) is essentially supported in a δ neighbourhood of S n−1 , and that we may assume g is constant at scale δ.
The appropriate maximal function in this setting is the Kakeya maximal function which is defined as follows: let N ≫ 1 be a large parameter, and for f defined on R n and ω ∈ S n−1 let where the sup is taken over all tubes T of dimensions 1 × · · · × 1 × N whose axis is parallel to ω. It is conjectured that 1 Strictly speaking (1) is equivalent to the variant of (4) where we average over spheres of radius r ∈ [1 − δ, 1 + δ] on both sides, ensuring that we may assume that the integrand on the right hand side is indeed essentially supported in B(0, δ −1 ). Without this interpretation it is not clear what meaning (4) may have in general. However, as we shall see below, we shall actually use (4) only in the case that g is constant at scale δ 1/2 , in which case there is no ambiguity. and this is also known to be true in two dimensions by work of Fefferman [11] but is open in higher dimensions. Once again, (4) and (5) fit into a standard machine which can be used to establish, inter alia, restriction estimates such as . .
Then (5) holds, that is and hence gdσ also holds.
The proof uses a two-scale technique and is similar to Fefferman's argument for the disc multiplier [10] and to the argument in [1] in which it was first noted that restriction estimates imply estimates for maximal functions, and which was later treated more formally by Bourgain [2]. Also see the remark at the end for another argument by-passing the maximal function.
Since we know that the Kakeya maximal conjecture implies the Nikodym maximal conjecture (see Theorem 4.10 of [14], which is in turn based upon [5]) we conclude that (4) and (1) (together) also imply the Bochner-Riesz conjecture (2).
Remark. This work was done in the early 1990's but was never presented for publication. The author has communicated its essence to a number of harmonic analysts over the years and its existence has apparently become known, resulting in occasional requests for it. It is hoped that this informal presentation will satisfy such demand.

Sketch of proof of Proposition 1
We shall establish (5) in a certain dual form, see for example [6] and [4]. For N fixed there are essentially N n−1 'distinct' directions that a 1 × · · · × 1 × N tube can occupy. Suppose we have a collection {T α } of such tubes, one in each direction. Then the best constant A r in the inequality (over all such families of tubes) is equivalent to the L r ′ → L r ′ operator norm of M N . The equivalent scaled version of (7) for families of λ × · · · × λ × λN tubes with distinct directions is α c α χ Tα r ≤ A r N 1/r λ n/r α |c α | r It is easy to verify that if the tubes {T α } form a bush (i.e. all pass through a common centre) then (7) holds for r = n/(n − 1) in the optimal form Proof. Warning: The treatment is somewhat informal.
Let T α be a collection of δ −1/2 × · · · × δ −1/2 × δ −1 tubes in R n (so λ = N = δ −1/2 here), one in each of the δ −(n−1)/2 directions. By the remarks above, it suffices to show the appropriately scaled form of (7) which is in this case We may assume that all the T α are supported in a ball B(0, δ −1 ) in R n and that T α is the translate by λ α of the tube passing through the origin in the α'th direction. Note that |λ α | δ −1 . We may assume that c α ≥ 0.
Let {E α } be δ 1/2 -cells on S n−1 and let where φ α is a smooth bump function associated to the cell E α . Standard stationary phase calculations give Since |λ α | δ −1 we have that g is roughly constant at scale δ and so we can further decompose g as g = β a β c where ψ β are smooth bump functions associated to a decomposition of the sphere into δ-cells F β , where α(β) is the α such that F β ⊆ E α and where |a β | ∼ 1.
Now we apply our assumption (4) -with δ 2 now playing the role of δ in (4) -to obtain .
Next we use Khintchine's inequality and (10) to deduce that Remark. Note that we did not need the full force of hypothesis (4) for general g, only for those g constant on δ 1/2 -cells.
Indeed, suppose we assume (4) for such g constant on δ 1/2 cells E α . We can then conclude inequality (6) directly without passing through the maximal function. To see this, we begin by observing that in order to prove inequality (6) we may assume that g is a step function on S n−1 which is constant at scale R −1 (this is the effect of restricting attention to B(0, R)). So, since B(0, R) ⊆ B(0, R 2 ), in order to prove (6) for such g it suffices to prove the ostensibly stronger 3 inequality for such g. But if we relabel R 2 as R, we see it therefore suffices to prove that inequality (6) holds for g constant at scale R −1/2 ; that is, we may assume g = α c α φ α . For such g we are assuming that we have inequality (4)  where the main contributions of the terms | φ α dσ| come from R 1/2 × · · · × R 1/2 × Rtubes centred at the origin, and which we may therefore calculate and estimate directly. Indeed, for such g constant at scale R −1/2 , we have 4 α |c α | 2 | φ α dσ| 2 1/2 L 2n n−1 (B(0,R)) where A t denotes a local average on the sphere at scale t. Now by the triangle inequality we have .