THE BOCHNER-RIESZ CONJECTURE IMPLIES THE RESTRICTION CONJECTURE

TERENCE TAO

1. Introduction

Fix n ≥ 2. For any 1 ≤ p ≤ ∞ and α ≥ 0 we use BR(p, α) to denote the statement that S^{δ(p)+α} is bounded on L^{p}, where δ(p) = max(n|\frac{1}{p} − \frac{1}{2}| − \frac{1}{2}, 0) and S^{δ} is the Bochner-Riesz multiplier

Ŝ^{δ}f(ξ) = (1 − |ξ|^{2})_{+}^{δ}f̂(ξ).

The Bochner-Riesz conjecture is the statement that BR(p, ε) holds for all 1 ≤ p ≤ ∞ and ε > 0. Apart from the trivial estimate BR(2, 0), this conjecture is optimal [13].

Similarly, we use R(p, α) to denote the localized restriction estimate^{1}

∥ℜf∥_{L^{p}(S^{n−1})} ≲ R^{α}∥f∥_{L^{p}(B(0,R))}

for f supported in B(0, R), where ℜf = f̂|_{S^{n−1}} is the sphere restriction operator. Note that R(p, 0) is equivalent to the global restriction estimate

∥ℜf∥_{L^{p}(S^{n−1})} ≲ ∥f∥_{L^{p}(R^{n})}.

The restriction conjecture asserts^{2} that R(p, 0) holds for all 1 ≤ p < 2n/(n + 1). Localized restriction theorems such as the L^{2} estimate R(2, 1/2) have been utilized before (see [1, 2]) as a stepping stone to global restriction theorems.

Although no formal equivalence has been proven between the two conjectures, the two conjectures are widely believed to be at least heuristically equivalent. For example, the implication Restriction ⟹ Bochner-Riesz is known for the parabolic analogue of the spherical problem (Carbery [4]), or when the (p, p) restriction hypothesis is strengthened to a (p, 2) estimate (see Fefferman[12], Christ [8, 7, 6], Tao[26], etc.). Figure 1 illustrates the close relationship between progress on the two conjectures.

1991 Mathematics Subject Classification. 42B15.

The author wishes to thank Tom Wolff for many valuable and insightful suggestions, and for indirectly motivating this paper. The author also thanks the reviewer for some helpful remarks.

1All of the estimates in this paper will be a priori, in the sense that all functions are assumed to be Schwarz class.

2There is a slight strengthening of this conjecture with imbalanced exponents, but we shall not concern ourselves with that here.

BR(p, ε) when  R(p, 0) when
p′ = ∞  Bochner  p′ = ∞  Riemann-Lebesgue
p′ ≥ 6  Fefferman, 1970 [12] p′ ≥ 6  Stein, 1967
p′ ≥ 4  Tomas, 1975 [28]  p′ > 4  Tomas, 1975 [28]
p′ ≥ 4  Stein, 1975
p′ ≥ 4 − \frac{8}{75}  Bourgain, 1991 [1]
p′ ≥ 4 − \frac{2}{15}  Bourgain, 1995 [2] p′ > 4 − \frac{2}{15} Bourgain, 1991 [1]
p′ ≥ 4 − \frac{2}{11}  Wolff, 1995 [30]  p′ > 4 − \frac{2}{11} Wolff, 1995 [30]
p′ = 3?  (critical value)  p′ > 3?  (critical value)

Figure 1. Progress on Bochner-Riesz and restriction for n = 3

In this note we present a formal proof of the implication in the reverse direction, that the Bochner-Riesz conjecture implies the restriction conjecture, as an immediate consequence of the following two results.

Theorem 1.1. If 1 ≤ p ≤ \frac{2n}{n+1}, then BR(p, α) ⟹ R(p, 2α).

Theorem 1.2. If 1 < p < 2 and 0 < α ≪ 1, then R(p, α) ⟹ R(q, 0) whenever

\frac{1}{q} > \frac{1}{p} + \frac{C}{log \frac{1}{α}} .

Corollary 1.3. If 1 < p ≤ \frac{2n}{n+1} and BR(p, ε) holds for all ε > 0, then R(q, 0) holds for all 1 ≤ q < p.

It is well-known that R(p, 0) is false for any p ≥ 2n/(n + 1), so the exponent in the above corollary is optimal.

The first theorem is based on a very simple observation, that the Bochner-Riesz operator resembles the restriction operator when evaluated at points far away from the support of the function. The factor of 2 in the conclusion is a familiar aspect of the numerology of the problem, and reflects the principle that the local behaviour at scale R is best studied at a global scale of R^{2}. A related “osculation” argument has appeared in [29].

The second theorem is more involved than the first, and can be viewed as a statement that if one can control the (L^{p}, L^{p}) norm of ℜ reasonably well on large balls, then one can control the (L^{q}, L^{p}) norm of the same operator on all of R^{n}, where q is a slightly worse exponent than p. The proof of the theorem requires two ideas; firstly, the observation that control of the restriction operator on large balls can be bootstrapped to control of the same operator on “sparse” unions of these large balls; and secondly, a “exponentially low-dimensional” Calderón-Zygmund decomposition which covers a set E by a small number of sparse collections of balls which are not too huge.

In both theorems the uncertainty principle plays a minor but recurring role. For example, we take advantage of the frequency localization of the restriction operator to introduce a spatial uncertainty of O(1), and conversely we use a spatial localization at scale R to introduce a frequency uncertainty of O(1/R). We also rely on the freedom to introduce an uncertainty of O(1) to the phase function of an oscillatory integral. One concrete statement of this principle is formalized as a stability lemma in Lemma 2.1.

Theorem 1.2 may be compared with a result of the same type which is due to Bourgain (see also [17]):

Theorem 1.4. [2] If p ≤ \frac{2(n+1)}{n+3}, 0 ≤ α ≤ \frac{n+1}{2p′}, then R(p, α) ⟹ R(q, 0) whenever

q′ ≥ 2 + \frac{1}{\frac{n+1}{2p′} − α}.

The theorem of Bourgain relies on a modification of the L^{2} restriction theory of Tomas and Stein [28], and is more efficient than Theorem 1.2 for most values of α; however for our purposes Theorem 1.2 is superior because it does not lose any exponents in the limit α → 0.

In the last section we present some extensions and related results to the main theorem; for instance, we show the equivalence of the Nikodym and Kakeya maximal operator conjectures. We will develop these results in a more general setting in a subsequent paper [27].

2. Proof of theorem 1.1

Let f be a compactly supported function, and fix p ≤ 2n/(n + 1), so that δ(p) = n(1/p − 1/2) − 1/2. The basic idea of the argument is to exploit the heuristic approximation

S^{δ(p)+α}f(x) ∼ |x|^{−α} \frac{e^{±2πi|x|}}{|x|^{n/p}} f̂(±\frac{x}{|x|})

which is accurate when x is very far away from the support of f. One may see this informally by using stationary phase to estimate the kernel of the Bochner-Riesz multiplier and then using a Taylor series to linearize the phase function.

We now begin the rigorous proof. By the standard Carleson-Sjölin reduction (see e.g. [25], [11], [5]), BR(p, α) implies the estimate

∥G_{R}f∥_{p} ≲ R^{−n/p′}R^{α}∥f∥_{p},

where

G_{R}f(x) = ∫ e^{2πiR|x−y|}ψ(x, y)f(y) dy,

and ψ(x, y) is a bump function supported on the set |x|, |y| ≲ 1, |x − y| ∼ 1. This estimate can be deduced from BR(p, α) by applying a partition of unity of the sphere to the multiplier for S^{δ(p)+α}, using the stationary phase asymptotics for the resulting kernel, localizing to the scale R and rescaling; see also the remarks at the end of this section.

If we replace R by R^{2} in the above estimate, perform the rescaling ỹ = y/R, and restrict x, ỹ to the range |x| ∼ 1, |ỹ| ≲ 1, we obtain

∥T_{R}g(x)∥_{p} ≲ R^{−n/p′}R^{2α}∥g∥_{p}  (1)

for all g supported in the unit ball, where

T_{R}g(x) = ∫ e^{2πi|R^{2}x−Rỹ|}ψ̃(x, ỹ)g(ỹ) dỹ

and ψ̃ is a bump function supported in the region |x| ∼ 1, |ỹ| ≲ 1.

We now use the Taylor series approximation

|R^{2}x − Rỹ| = R^{2}|x| − R⟨\frac{x}{|x|}, ỹ⟩ + a(x, ỹ, 1/R),

where a(x, ỹ, τ) is a bounded function which is C^{∞} in x and τ for |x| ∼ 1, 0 ≤ τ ≲ 1, |ỹ| ≲ 1. The a(x, ỹ, 1/R) error term in the phase can be dropped, as can the ỹ-dependence of the amplitude function ψ̃(x, ỹ) (see remarks at the end of this section), so that T_{R} can be replaced by T̃_{R} in (1), where

T̃_{R}g(x) = ψ(x)e^{2πiR^{2}|x|} ∫ e^{−2πiR⟨ \frac{x}{|x|} ,ỹ⟩}g(ỹ) dỹ = ψ(x)e^{2πiR^{2}|x|}ĝ(R\frac{x}{|x|}),

and ψ is a function supported on {|x| ∼ 1}. The conclusion R(p, 2α) then follows by rescaling.

We now justify the heuristic principle, used repeatedly in the above argument, that one can freely modify the amplitude or phase of a compactly supported oscillatory integral by a smooth O(1) factor without affecting its regularity properties. Variations of this stability lemma have appeared elsewhere in the literature, see e.g. Lemma 2.10 of Christ [9], or the argument in [16] (see also [2], [23]).

Lemma 2.1. For each R ≫ 1, let K_{R}(x, y) be a bounded, compactly supported function, where the bounds and support are independent of R, and let 1 ≤ p, q ≤ ∞. Suppose b(x, y, τ) is a bounded function which is C^{∞} in x and τ, for 0 ≤ τ ≲ 1 and x, y in the support of K. Then, if the operators A_{R} and B_{R} are defined by

A_{R}f(x) = ∫ K_{R}(x, y)f(y) dy

B_{R}f(x) = ∫ K_{R}(x, y)b(x, y, 1/R)f(y) dy,

then for all N > 0 we have

∥B_{R}∥_{L^{p}→L^{q}} ≲ ∥A_{R}∥_{L^{p}→L^{q}} + O(R^{−N}).

In particular, if b = e^{ia} for some a(x, y, τ) then the operator norms of A_{R} and B_{R} are comparable (except for a rapidly decreasing error).

Proof. Since x, y are compactly supported, one can break b up into a Fourier series in x and a Taylor series in τ as

b(x, y, τ) = ∑_{j=0}^{N} ∑_{k∈Z^{n}} c_{j,k}τ^{j}e^{2πi⟨k,x⟩}f_{j,k}(y) + O(τ^{N}),

where the f_{j,k} are uniformly bounded functions, the c_{j,k} are constants which are rapidly decreasing in k, Z^{n} is a suitable lattice, and N is an arbitrarily large number. The result then follows from the operator identity

B_{R} = ∑_{j=0}^{N} ∑_{k∈Z^{n}} c_{j,k}R^{−j}e^{2πi⟨k,·⟩}A_{R}f_{j,k} + O(R^{−N})

and the triangle inequality.

Note that the L^{p} spaces in the above lemma can easily be replaced by Lorentz spaces, Orlicz spaces, or any other quasi-normed spaces which are stable under multiplication by L^{∞} functions. The requirement that K is compactly supported in y can also be weakened. The fact that no regularity is assumed in y, although not needed in the proof of the above theorem, is useful in other applications, such as replacing a continuous estimate by its discrete analogue or vice versa. (See e.g. Bourgain[2] for examples of such heuristics).

3. Proof of theorem 1.2

The first step is to bootstrap the localized restriction estimate so that it applies to functions which are supported on a sparse union of balls of constant radius. The idea is to exploit the fact that the Fourier transforms of functions on widely seperated balls behave in a quasi-orthogonal manner on the unit sphere.

Definition 3.1. A collection {B(x_{i}, R)}_{i=1}^{N} of R-balls is sparse if the centers x_{i} are R^{C}N^{C} seperated. (C denotes a constant that will change from line to line).

Lemma 3.2. Suppose R(p, α) holds for some α > 0 and 1 < p < 2. Then

∥ℜf∥_{p} ≲ R^{α}∥f∥_{p}

whenever f is supported on ⋃_{i} B(x_{i}, R) and {B(x_{i}, R)} is a sparse collection of balls.

Proof. We start by modifying the localized restriction hypothesis slightly. Let ℜ̃f be the restriction of f̂ to the annulus A_{R} of thickness ∼ 1/R around the unit sphere. By averaging the restriction operator over all dilations with dilation factor 1 + O(1/R), we see that R(p, α) implies the estimate

∥ℜ̃f∥_{L^{p}(A_{R})} ≲ R^{−1/p}R^{α}∥f∥_{p}  (2)

whenever f is supported on B(0, R). By translation symmetry, we see that this estimate is also valid if f is supported in any other ball B(x, R) of radius R.

Write f = ∑_{i} f_{i}φ_{i}, where each f_{i} is supported on B(x_{i}, R), φ_{i}(x) = φ(\frac{x−x_{i}}{R}), and φ is a Schwartz function which is positive on the unit ball, and whose Fourier

6 TERENCE TAO

transform is supported on the unit ball. Since ∥f∥_{p} ∼ (∑_{i} ∥f_{i}∥_{p}^{p})^{1/p} and ℜf = ∑_{i} ℜ̃f_{i} ∗ φ̂_{i}|_{S^{n−1}}, the lemma will follow from (2) if we can prove the estimate

∥ ∑_{i} F_{i} ∗ φ̂_{i}|_{S^{n−1}}∥_{p} ≲ R^{1/p}(∑_{i} ∥F_{i}∥_{p}^{p})^{1/p}

for all F_{i} ∈ L^{p}(R^{n}). When p = 1 this estimate follows from replacing everything by absolute values and using Fubini’s theorem, so it suffices by real interpolation to prove the estimate for p = 2. By Plancherel’s theorem this estimate is equivalent to the inequality

∥ℜ(∑_{i} f_{i}φ_{i})∥_{2} ≲ R^{1/2}(∑_{i} ∥f_{i}∥_{2}^{2})^{1/2}

for all f_{i} ∈ L^{2}(R^{n}). By the T^{∗}T method we can then replace the inequality with the equivalent inequality

(∑_{j} ∥φ_{j}ℜ^{∗}ℜ(∑_{i} f_{i}φ_{i})∥_{2}^{2})^{1/2} ≲ R(∑_{i} ∥f_{i}∥_{2}^{2})^{1/2}

By Schur’s test and self-adjointness this estimate will follow from

sup_{j} ∑_{i} ∥φ_{j}ℜ^{∗}ℜφ_{i}∥ ≲ R,

where ∥T ∥ denotes the L^{2} operator norm of T. This in turn will follow from the estimates

∥φ_{i}ℜ^{∗}ℜφ_{i}∥ ≲ R

and

∥φ_{j}ℜ^{∗}ℜφ_{i}∥ ≲ R^{−C}N^{−C}, j ≠ i.

To prove the former estimate^{3} we use Plancherel’s theorem to rewrite the estimate as

∥φ̂_{i} ∗ (dσ(φ̂_{i} ∗ g))∥_{2} ≲ R∥g∥_{2},

where dσ is surface measure on the sphere. But this follows from the trivial estimate

∥φ̂_{i} ∗ (dσ(φ̂_{i} ∗ g))∥_{∞} ≲ R∥g∥_{∞},

self-adjointness, and interpolation (or Schur’s test). To prove the latter estimate, we rewrite it as

∥φ_{j}(d̂σ ∗ (φ_{i}f))∥_{2} ≲ R^{−C}N^{−C}∥f∥_{2}.

By self-adjointness and interpolation it suffices to show that

∥φ_{j}(d̂σ ∗ (φ_{i}f))∥_{∞} ≲ R^{−C}N^{−C}∥f∥_{∞}.

But this follows from the fact that φ_{i} and φ_{j} are rapidly decreasing away from x_{i} and x_{j} respectively, and that the size of d̂σ evaluated near x_{j} − x_{i} is O(|x_{j} − x_{i}|^{−(n−1)/2}) = O(R^{−C}N^{−C}).

3One can also obtain this estimate from the localized restriction theorem R(2, 1/2); see e.g.

[2], [17].

Fix p < 2 and α > 0, and assume that R(p, α) holds. To prove the global restriction theorem we shall start with some standard reductions. Firstly, it suffices to show the Lorentz space estimate

∥ℜf∥_{p} ≲ ∥f∥_{q_{0},1},  (3)

where 1/q_{0} = 1/p + C/ log(1/α); this will imply the theorem by Hölder’s inequality and interpolation with known restriction theorems such as the trivial (L^{1}, L^{∞}) estimate.

By averaging over translations^{4} it suffices to show (3) when f is a measure supported on a discrete lattice Z^{n}, and the L^{q_{0},1} norm is replaced by the discrete norm l^{q_{0},1}. We may then replace f by f ∗ χ (and revert to the continuous norm L^{q_{0},1}), where χ is the characteristic function of the cube of size c, and c ∼ 1 is chosen so that χ̂ is positive on the unit sphere. Combining the two reductions we see that it suffices to verify the original estimate (3) when f is constant on c-cubes.

Since we are working in L^{q_{0},1} we may take f = χ_{E} for some set E, which we can assume to be the union of c-cubes. Our objective is now to prove that

∥ℜχ_{E}∥_{p} ≤ C_{α}|E|^{1/p+C/ log(1/α)}  (4)

This will be accomplished with the aid of the following Calderón-Zygmund type lemma, which covers such a set E by a reasonably small number of sparse collections of balls, where one has some modest control on the size of the balls.

Lemma 3.3. Suppose E is the union of c-cubes, and N ≥ 1. Then there exist O(N|E|^{1/N}) sparse collections of balls which cover E, such that the balls in each collection have radius O(|E|^{C^{N}}).

Proof. Define the radii R_{k} for 0 ≤ k ≤ N by R_{0} = 1, R_{k+1} = |E|^{C}R_{k}^{C}; note that R_{k} = O(|E|^{C^{k}}) for each k.

Starting with k = 1 and proceeding recursively, we set E_{k} to be the set of all x ∈ E which are not in any E_{j} for j < k and are such that

|E ∩ B(x, R_{k})| ≤ |E|^{k/N}.

For every 1 ≤ k ≤ N and x ∈ E_{k} we have by construction and hypothesis that

|E ∩ B(x, R_{k−1})| ≳ |E|^{(k−1)/N}.

Thus for every x ∈ E_{k} the set E_{k} ∩ B(x, R_{k}) can be covered by O(|E|^{1/N}) R_{k−1}- balls. This implies that the entire set E_{k} can be covered by O(|E|^{1/N}) collections of R_{k−1}-balls which are R_{k}-seperated. Since the cardinality of these collections can be at most O(|E|) we see from the construction of the R_{k} that the collections are sparse. The lemma then follows by noting that E = ⋃_{k=1}^{N} E_{k}.

We can now conclude the proof of Theorem 1.2. If E is the union of c-cubes, then by Lemma 3.3 we may cover E with O(N|E|^{1/N}) sets E_{j} which are each the union of a sparse collection of balls of radius O(|E|^{C^{N}}); by the Vitali covering lemma we may assume that |E_{j}| ≲ |E|. Applying Lemma 3.2 to each such E_{j} we obtain

4For a proof of the fact that L^{q_{0},1} is a Banach space, see [24].

∥ℜχ_{E_{j}}∥_{p} ≲ (|E|^{C^{N}})^{α}|E|^{1/p}.

By the triangle inequality this yields

∥ℜχ_{E}∥_{p} ≲ N|E|^{1/N}(|E|^{C^{N}})^{α}|E|^{1/p},

and (4) follows by setting N = C^{−1} log(1/α) for some sufficiently large C.

4. Additional remarks

Several variations on the above theme are possible. Further results on these problems will be presented in future papers (e.g. [27]).

In this section we shall discuss some open conjectures related to spherical Bochner-Riesz and spherical restriction conjectures, which we describe briefly below for reference. (The notation and wording of the conjecture may differ slightly from the standard formulation).

Conjecture 4.1. (Local smoothing for the wave equation [22]) If u(x, t) is a solution of the homogeneous Cauchy problem

(−\frac{∂^{2}}{∂t^{2}} + Δ)u(x, t) = 0

u(x, 0) = f(x), u_{t}(x, 0) = 0,

then one has

∥u∥_{L^{p}(R^{n}×[1,2])} ≲ ∥(1 + \sqrt{−Δ})^{ε}f∥_{p}

for all ε > 0, where p = 2n/(n − 1) and Δ is the Laplacian.

Conjecture 4.2. (Maximal spherical Bochner-Riesz^{5} for p ≥ 2 [3], [6]) For R, δ > 0 define the operators S_{R}^{δ} by

Ŝ_{R}^{δ}f(ξ) = (1 − |\frac{ξ}{R}|^{2})_{+}^{δ}f̂(ξ).

Then for all 2 ≤ p ≤ ∞ and δ > δ(p), one has the estimate

∥ sup_{R>0} S_{R}^{δ}f∥_{p} ≲ ∥f∥_{p}.

Conjecture 4.3. (Parabolic Bochner-Riesz [4]) For δ > 0 define the operators S̃^{δ} by

ˆS̃_{R}^{δ}f(\underline{ξ}, ξ_{n}) = η(ξ)(ξ_{n} − \frac{1}{2}|\underline{ξ}|^{2})_{+}^{δ}f̂(\underline{ξ}, ξ_{n}),

for (\underline{ξ}, ξ_{n}) ∈ R^{n−1} × R, where η is smooth and compactly supported. Then S̃^{δ} is bounded on L^{p} whenever 1 ≤ p ≤ ∞ and δ > δ(p).

Conjecture 4.4. (Parabolic restriction [4], [17]) Let S = \{(\underline{ξ}, \frac{1}{2}|\underline{ξ}|^{2}) : |\underline{ξ}| ≲ 1\} be a subset of the standard paraboloid. Then for all 1 ≤ p < 2n/(n − 1) we have

∥f̂|_{S}∥_{L^{p}(S)} ≲ ∥f∥_{p}.

5The conjecture is false for some p < 2; see [27].

Conjecture 4.5. (Kakeya maximal function [1]) If f is a compactly supported function and 0 < δ ≪ 1, we define the Kakeya maximal function f_{δ}^{∗} on S^{n−1} by

f_{δ}^{∗}(ω) = sup_{T//ω} \frac{1}{|T |} ∫_{T} f(x) dx,

where the supremum is taken over all 1 × δ tubes T which are oriented in the direction ω. Let K(p, α) denote the estimate

∥f_{δ}^{∗}∥_{p} ≲ δ^{\frac{n}{p} −1−α}∥f∥_{p}.

Then K(p, α) holds for all 1 ≤ p ≤ n and α > 0.

Conjecture 4.6. (Kakeya set [1]) Define a Kakeya set to be any subset E of R^{n} which contains a unit line segment in each direction. Then all Kakeya sets have Minkowski dimension n.

Conjecture 4.7. (Nikodym maximal function [1]) If f is a compactly supported function and 0 < δ ≪ 1, we define the Kakeya maximal function f_{δ}^{∗} on R^{n} by

f_{δ}^{∗}(x) = sup_{x∈T} \frac{1}{|T |} ∫_{T} f(y) dy,

where the supremum is taken over all 1 × δ tubes T which contain x. Let N(p, α) denote the estimate

∥f_{δ}^{∗∗}∥_{p} ≲ δ^{\frac{n}{p} −1−α}∥f∥_{p}

Then N(p, α) holds for all 1 ≤ p ≤ n and α > 0.

Conjecture 4.8. (Nikodym set [1]) Define a Nikodym set to be any subset E of R^{n} such that for every x ∈ R^{n} there exists a line l through x such that E ∩ l contains a unit line segment. Then all Nikodym sets have Minkowski dimension n.

It is known ([14], [11]) that both the spherical and parabolic restriction conjectures imply the Kakeya maximal conjecture, and that [1] the Kakeya (resp. Nikodym) maximal function conjecture implies the Kakeya (resp. Nikodym) set conjecture. Also, the local smoothing conjecture of Sogge [22] is known to imply the maximal Bochner-Riesz conjecture and hence the Bochner-Riesz conjecture. Finally, the parabolic restriction conjecture is known to imply the parabolic Bochner-Riesz conjecture [4]. These equivalences, together with the ones discussed in this paper, are summarized in Figure 2.

The arguments of Sections 2 and 3 can be generalized to arbitrary oscillatory integrals. Informally, the result is that any oscillatory integral estimate can be linearized into a global restriction theorem. More precisely, suppose

S_{R}f(x) = ∫ e^{−2πiRϕ(x,y)}ψ(x, y)f(y) dy,

is a family of oscillatory integrals, where the amplitude ψ is smooth and compactly supported in R^{n} ×R^{m}, where m−1 ≤ n ≤ m, and ϕ is a phase function of constant rank m−1 in the sense of Hörmander [15]. Then for each point y_{0} in the domain of y one can associate a Fourier restriction operator ℜ_{y_{0}} associated to the linearization of the S_{R} at y_{0}:

ℜ_{y_{0}}f(x) = ψ(x, y_{0})f̂(∇_{y}ϕ(x, y_{0})).

local smoothing ⇓ maximal Bochner-Riesz ⇓ spherical Bochner-Riesz ⇓ spherical restriction ⇓

parabolic restriction ⟺ parabolic Bochner-Riesz
⇓
maximal Kakeya ⟺ maximal Nikodym
⇓

Kakeya set ⇓ Nikodym set

Figure 2. Known implications between Bochner-Riesz, Kakeya, and related conjectures.

Theorem 4.9. If S_{R} is bounded from L^{p}(R^{m}) to L^{p}(R^{n}) with operator norm O_{ε}(R^{−m/p′+ε}) for all ε > 0, then for each y_{0} and all 1 ≤ q < p, the operators ℜ_{y_{0}} are bounded from L^{q}(R^{m}) to L^{q}(R^{n}).

The proof of the theorem are routine modifications of the arguments given in Sections 2 and 3. Note that the hypothesis of Theorem 4.2 implies that the restriction surface {∇_{y}ϕ(x, y_{0}) : x ∈ R^{n}} cannot degenerate to infinite order because of the Knapp counterexample. Hence one can use stationary phase to obtain some decay to Fourier transforms of measures on this surface, so that the argument in Lemma 3.1 carries over.

As an example of this theorem we may reverse the results of Carbery[4] and conclude that the Bochner-Riesz and restriction conjectures for paraboloids are equivalent. Another example (see [18]) comes from the Bochner-Riesz oscillatory integrals S_{λ} arising from a Riemannian manifold M^{n}:

S_{λ}f(x) = ∫_{M^{n}} e^{iλdist(x,y)}a(x, y)f(y) dy.

Any positive (L^{p}, L^{p}) result on the Hörmander problem for this operator implies a corresponding result for the spherical restriction operator, since the linearization of the distance function is a equivalent after a linear change of variables to the spherical restriction phase function.

A variation of the above methods shows that the full spherical restriction conjecture implies the full parabolic restriction conjecture. The key observation is that the restriction theorem at the critical index p = 2n/(n + 1) is invariant under parabolic scaling. Thus, one can scale the spherical restriction conjecture at this index until the sphere resembles a paraboloid. (Lemma 2.1 can be used to make the notion of “resembling” precise). Of course, one does not have the restriction theorem at the endpoint index, so that in practice one only gets the localized restriction theorem R(p, ε) for paraboloids for all p < 2n/(n + 1) and ε > 0. One then removes the epsilon by the techniques in Section 3. We omit the details.

Our final result is that the Kakeya and Nikodym maximal function conjectures are equivalent.

Theorem 4.10. If 1 ≤ p ≤ n and 0 ≤ α < (n + 1)/p, then K(p, α) ⟹ N(p, α) ⟹ K(p, 2α).

Proof. The latter implication is an analogue of Theorem 1.2, and follows immediately from the pointwise estimate

δf_{δ}^{∗}(\frac{x}{|x|}) ≲ (f_{δ})_{Cδ^{2}}^{∗∗}(x)

where f_{δ}(x) = f(x/δ), f is a compactly supported function, and |x| ∼ 1. This estimate reflects the geometric fact that the 1 × δ^{2} tubes through x that intersect the support of f_{δ} are essentially rescaled versions of 1 × δ tubes oriented in the direction x/|x|.

The former implication is an analogue of the argument in [4]. The idea is most easily expressed for the set conjecture as the observation that a Nikodym set can always be mapped onto a Kakeya set by a projective transformation. To prove the corresponding implication for the maximal operators we first note that to prove N(p, α) it suffices to show the frozen estimate

∥f_{δ}^{∗∗}(\underline{x}, 0)∥_{p} ≲ δ^{\frac{n}{p} −1−α}∥f∥_{p},

where we parameterize R^{n} by x = (\underline{x}, x_{n}). We may assume f is supported on the slab 0 < x_{n} ≤ 1. It suffices to prove the estimate for the slab 1/2 < x_{n} ≤ 1, since the condition α < (n + 1)/p and scaling considerations ensure that the other contributions are more favourable than this main term.

The implication is now immediate from the pointwise estimate

f_{δ}^{∗∗}(\underline{x}, 0) ≲ (f ∘ ϕ)_{Cδ}^{∗}(\frac{(\underline{x}, 1)}{|(\underline{x}, 1)|}),

where

ϕ(\underline{x}, x_{n}) = (\frac{\underline{x}}{x_{n}} , \frac{1}{x_{n}})

is a projective transformation. This reflects the geometric fact that tubes through the point (\underline{x}, 0) on the slab \{1/2 < x_{n} ≤ 1\} are transformed under ϕ to tubes in the direction \frac{(\underline{x},1)}{|(\underline{x},1)|}.

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Department of Mathematics, UCLA, Los Angeles CA 90095-1596

E-mail address: tao@math.ucla.edu