THE WEAK-TYPE ENDPOINT BOCHNER-RIESZ CONJECTURE AND RELATED TOPICS

TERENCE TAO

Abstract. We continue the study of the Bochner-Riesz family of conjectures. In particular, the weak-type endpoint Bochner-Riesz conjecture is shown to be equivalent to the standard Bochner-Riesz conjecture, and the maximal Bochner-Riesz conjecture is shown to fail for certain p < 2. A new Radon transform conjecture is also formulated.

1. Introduction

We continue the study initiated in [33], in which several implications between the Bochner-Riesz family of conjectures were proven.

In Sections 3 and 4 we show that the Bochner-Riesz conjecture is equivalent to its weak-type endpoint version, with only an epsilon loss in exponents. As a corollary one obtains the weak-type endpoint boundedness of the Bochner-Riesz multipliers on L^{p} for p < 42/31 in three dimensions, improving on the results in [32]. The proof combines the linearization and sparse decomposition introduced in [33] with the reduction to strong-type estimates employed in [25]. As a corollary we obtain an epsilon-removal result for the Hörmander oscillatory integral associated to the Bochner-Riesz multipliers.

Since Stein’s maximal principle [28] relates maximal operators to weak-type estimates for p ≤ 2, it is natural to consider the boundedness of the maximal Bochner-Riesz operators in this range. Surprisingly, a simple example shows that these operators are not bounded on the naive range of exponents, even when n = 2, thus giving a negative answer to a question posed in [29]. We shall discuss this phenomenon in Section 5.

In the Section 6 we show how the local smoothing conjecture of Sogge [27], together with the closely related square-function spherical Bochner-Riesz conjecture, can be linearized in the scattering limit to form a new Radon transform estimate, which in turn implies the spherical restriction conjecture.

The author wishes to thank Tom Wolff, Andreas Seeger, Bill Minicozzi, Michael Christ, and Chris Sogge for many stimulating and helpful discussions on this subject. This work was supported by NSF grant DMS-9706764.

2. Notation

In this section we set out the notation which is common to all the sections of this paper.

1991 Mathematics Subject Classification. 42B15.

The dimension n will always be an integer greater than 1. We will occasionally need to parameterise a point x ∈ R^{n} by

x = (\underline{x}, x_{n}) = \underline{x} + x_{n}e_{n},

where x_{n} ∈ R and \underline{x} ∈ R^{n−1} is orthogonal to the unit basis vector e_{n}.

We use A ≲ B to denote the estimate A ≤ CB, where C is a positive constant that depends only on n and such parameters as the exponent p, the indices δ, α, and an epsilon ε. We say that A ∼ B if A ≲ B and B ≲ A.

If m(ξ) is a function, we use m(D) to denote the corresponding Fourier multiplier

m̂(D)f(ξ) = m(ξ)f̂(ξ);

D can be interpreted as the momentum operator D = \frac{1}{2πi}∇. For each δ ≥ 0 we form the Bochner-Riesz multiplier S̃^{δ} = m̃^{δ}(D), where

m̃^{δ}(ξ) = (1 − |ξ|^{2})_{+}^{δ}.

For technical reasons it will be convenient to replace this operator by S^{δ} = m^{δ}(D), where

m^{δ}(ξ) = (1 − |ξ|^{2})_{+}^{δ}φ(ξ)

and φ(ξ) is a bump function supported on |ξ − e_{n}| ≤ 1/4. Since S̃^{δ} can be reconstructed from a finite number of rotated copies of S^{δ} and some smoothing operators, any estimate on S^{δ} implies the corresponding one for S̃^{δ}. From standard stationary phase estimates we see that the convolution kernel K^{δ} of S^{δ} is given by

K^{δ}(x) = (1 + |x|)^{− \frac{n+1}{2} −δ}e^{±2πi|x|}a(x) + smoothing error,  (1)

where a is a standard symbol of order 0 which vanishes near the hyperplane x_{n} = 0, and the symbol ± represents +1 if x_{n} > 0 and −1 otherwise.

More generally, for any t > 0 we define the multipliers S̃_{t}^{δ}, S_{t}^{δ} by

S̃_{t}^{δ} = m̃^{δ}(D/t), S_{t}^{δ} = m^{δ}(D/t).

As is usual in these problems, we will need a dyadic decomposition of the Bochner-Riesz multiplier m^{δ(p)}. It will be convenient to alternate between a frequencylocalized decomposition and a spatially-localized decomposition, and so we will define several types of multipliers.

Definition 2.1. For each j > 0, we say that a multiplier m_{j} is type j if one has

|∂^{α}m_{j}(ξ)| ≲ C_{α,N}2^{j|α|}(1 + 2^{j} |1 − |ξ||)^{−N}

for all α, N ≥ 0. If m_{j} is supported on the double annulus |1 − |ξ|| ∼ 2^{−j}, then we say that m_{j} is frequency-localized type j. If m̂_{j} is supported on the ball |x| ≲ 2^{j}, we say that m_{j} is weakly spatially-localized of type j. If m̂_{j} is supported on the annulus |x| ∼ 2^{j}, we say that m_{j} is strongly spatially-localized type j.

When j ≤ 0 we adopt the convention that the only type j multiplier is the zero multiplier m_{j} = 0.

We will use two dyadic decompositions of the Bochner-Riesz multipliers m^{δ}. Firstly, we may find multipliers m_{j} of frequency-localized type j for each j such that

m^{δ(p)}(D) = ∑_{j} 2^{−jδ(p)}m_{j}(D);  (2)

see [30], [25], [10, 9]. This type of decomposition is especially good for studying the weak-type endpoint problem. We will also need the decomposition

m^{δ(p)}(D) = ϕ(D) + ∑_{j} 2^{−jδ(p)}m_{j}(D),  (3)

where the multipliers m_{j} are strongly spatially-localized type j and ϕ(D) is a harmless smoothing operator. See [31], [15], [30].

3. Weak-type endpoint bounds for Bochner-Riesz means: preliminaries

The Bochner-Riesz conjecture states that S^{δ(p)+α} is bounded on L^{p} for all α > 0 and 1 ≤ p ≤ ∞, where δ(p) = max(n|\frac{1}{p} − \frac{1}{2}| − \frac{1}{2}, 0). Apart from the trivial observation that S^{0} is bounded on L^{2}, this conjecture is optimal [16]. In this section we shall restrict our attention to the case 1 ≤ p ≤ 2n/(n + 1), so that δ(p) = n/p − (n + 1)/2; the full Bochner-Riesz conjecture can be obtained from this case by duality and interpolation.

This conjecture was proven in two dimensions by Carleson and Sjölin [6], but is still open in higher dimensions. For n = 3 the best result known is due to Wolff [35], who proved the conjecture for p ≤ 42/31.

Closely related to the Bochner-Riesz conjecture is the restriction conjecture, which states that^{1}

∥f̂|_{S^{n−1}}∥_{p} ≲ ∥f∥_{p}  (4)

for all 1 ≤ p < 2n/(n + 1). Progress on the restriction conjecture has closely paralleled the Bochner-Riesz conjecture; in fact, we have the implication

Theorem 3.1. [33] If the Bochner-Riesz conjecture is true for p = p_{0}, where 1 ≤ p_{0} ≤ 2n/(n + 1), then the restriction conjecture is true for all 1 ≤ p < p_{0}.

A natural endpoint version of the Bochner-Riesz conjecture is the statement that that S^{δ(p)} is of weak-type (p, p), for 1 ≤ p < 2n/(n + 1). For 2n/(n + 1) ≤ p ≤ 2n/(n − 1) no such estimate is possible, from the results in [16]. Also, no estimate is possible when p > 2n/(n − 1); by examining S^{δ(p)} on a small bump function we see that S^{δ(p)} does not map the Lorentz space L^{p′,1} to L^{p′}, and the weak-type estimate fails by duality.

In two dimensions the full endpoint conjecture was proven by Seeger [25]. In higher dimensions the result was proven for p < 2(n + 1)/(n + 3) by Christ [10, 9], and extended to p = 2(n + 1)/(n + 3) by the author [32]. For radial functions the conjecture was verified by [13].

Theorem 3.2. If the Bochner-Riesz and restriction conjectures are true for p = p_{0}, where 1 ≤ p_{0} ≤ 2n/(n + 1), then S^{δ(p)} is of weak-type (p, p) for all 1 ≤ p < p_{0}.

By Theorem 3.1 the restriction theorem hypothesis can be removed. Since the Bochner-Riesz conjecture follows from its endpoint version by duality and interpolation (see [15]), we see that the Bochner-Riesz conjecture is thus equivalent to its weak-type endpoint version. In the concrete situation n = 3, we obtain that S^{δ(p)} is of weak-type (p, p) for all 1 ≤ p < 42/31, improving on the results of [32].

1All estimates in this paper will be a priori, in that they are only assumed to hold for Schwarz functions.

The proof is somewhat involved, combining the quasi-orthogonality techniques of [25] with the sparse Calderón-Zymgund decomposition of [33]. As in [33], the uncertainty principle plays a recurring role, allowing us to co-ordinate the frequency space behaviour of the multiplier with the physical space structure of the function. Proof By (2), the weak (p, p) boundedness of S^{δ(p)} will follow from the estimate

∥ ∑_{j} 2^{−jδ(p)}m_{j}(D)f∥_{p,∞} ≲ ∥f∥_{p}  (5)

for all frequency-localized type j multipliers m_{j}.

On the other hand, from the Bochner-Riesz hypothesis we have

∥m_{j}(D)f∥_{p_{0}} ≲ 2^{njδ(p_{0})}2^{εj}∥f∥_{p_{0}}  (6)

for all ε > 0 and all type j multipliers m_{j}. To see this, we use the standard Carleson-Sjölin-Hörmander reduction (see e.g. [6], [30], [2]) to reduce this estimate to

∥ ∫ e^{2πi2^{j}|x−y|}a(x, y)f(y) dy∥_{p_{0}} ≲ 2^{−_{\frac{j}{p′_{0}}}} 2^{εj}∥f∥_{p_{0}}  (7)

for all j > 0 and all smooth cutoff functions a(x, y) which are supported away from the diagonal x = y. It suffices to prove this for one such a(x, y), since by translation, rotation, and dilation symmetry one can make the support of a(x, y) arbitrarily large, and by the stability lemma^{2} one can multiply a(x, y) by an arbitrary smooth compactly supported function.

However if one localizes the Bochner-Riesz kernel to a region where |x − y| ∼ 2^{j}, |x|, |y| ∼ 2^{j}, applies (1) and then rescales by 2^{j}, one sees that the hypothesis on the Bochner-Riesz multiplier does indeed imply an estimate of the form (7). Thus the estimate (6) holds.

It remains to obtain (5) from (6) and the restriction hypothesis. Our first step will be to make a convenient reduction, which exploits the uncertainty principle to make f constant on small cubes.

Lemma 3.3. There exists a dyadic number 0 < c ≪ 1 such that every function f ∈ L^{p}, 1 ≤ p ≤ ∞, can be decompsed as f = f_{1} + f_{2}, where ∥f_{1}∥_{p}, ∥f_{2}∥_{p} ≲ ∥f∥_{p}, f_{1} is constant on dyadic c-cubes, and the Fourier transform of f_{2} vanishes on {ξ : |ξ| < 2}.

Proof For any f ∈ L^{p}, let E(f) be the orthogonal projection of f onto the space of functions which are constant on dyadic c-cubes; note that ∥E(f)∥_{p} ≲ ∥f∥_{p}. Let ψ be a bump function such that ψ(ξ) = 1 when |ξ| < 2. We have

f = E(f) + ψ(D)(f − E(f)) + (1 − ψ(D))(f − E(f)).

The first and second terms can be absorbed into f_{1} and f_{2} respectively. As for the last term, one can use integration by parts to obtain the estimate

∥(1 − ψ(D))(f − E(f))∥_{p} ≤ \frac{1}{2}∥f∥_{p}

for all 1 ≤ p ≤ ∞ if c is sufficiently small, and so an iteration argument thus gives the desired decomposition.

2See Lemma 2.1 of [33], or Lemma 2.10 of [11]; the idea is to first consider the case when the multiplier function is of the form e^{2πi(k·x+l·y)}, and use Fourier series to handle the general case.

From the lemma and the fact that ∑_{j} 2^{−nδ(p)}m_{j} is supported on |ξ| < 2 we see that it suffices to prove (5) for functions f which are constant on dyadic c-cubes. (See [15] for a related argument).

The heart of the proof lies in the following quasi-orthogonality estimate, of the type first introduced by Seeger [25].

Proposition 3.4. If (6) and the restriction hypothesis hold, then

∥ ∑_{j} 2^{−jδ(p)}m_{j}(D)f_{j}∥_{p} ≲ (∑_{j} ∥f_{j}∥_{p}^{p})^{1/p}  (8)

for all functions f_{j} which are constant on c-cubes, all 1 < p < p_{0}, and all frequencylocalized type j multipliers m_{j}.

We will defer the proof of this proposition to the next section, preferring to close this section instead with the derivation of (5) from the proposition.

We follow the arguments of [25], which are in turn based on the work in [15] and the subsequent refinements in [10], [9]. It suffices by linearity to show that

|{| ∑_{j} 2^{−jδ(p)}m_{j}(D)f| ≳ 1}| ≲ ∥f∥_{p}^{p}.

We apply the Calderón-Zygmund decomposition to |f|^{p} at level 1 to write f = g + ∑_{I} b_{I}, where the function g satisfies ∥g∥_{∞} ≲ 1, ∥g∥_{p} ≲ ∥f∥_{p}, and the functions b_{I} are supported on disjoint dyadic cubes I and satisfy ∥b_{I}∥ ≲ |I|^{1/p} and ∑_{I} |I| ≲ ∥f∥_{p}^{p}. (See [15]). Note that by the assumption on f, all the cubes I have sidelength at least c. After disposing of the “good” function g in the usual manner (e.g. using L^{2} estimates and Plancherel’s theorem, as in [15]), it suffices to show that

|{| ∑_{j} ∑_{I} 2^{−δ(p)j}m_{j}(D)b_{I}| ≳ 1}| ≲ ∑_{I} |I|.

Let 2^{i} denote the sidelength l(I) of I. From Chebyshev’s inequality and the triangle inequality the above distributional estimate will follow from the strong-type estimate

∥ ∑_{I} 2^{−δ(p)(i+s)}m_{i+s}(D)b_{I}∥_{L^{q}(R^{n}\ ⋃_{I} CI)} ≲ 2^{−ε|s|}(∑_{I} |I|)^{1/q}  (9)

for all s, where ε > 0 and q will be chosen later, and CI denotes a dilate of I. In contrast, if one sets f_{i+s} = ∑_{l(I)=2^{i}} b_{I} in (8), one obtains

∥ ∑_{I} 2^{−δ(p)(i+s)}m_{i+s}(D)b_{I}∥_{p} ≲ (∑_{I} |I|)^{1/p}  (10)

for all s. Thus for s > 0 (9) can be obtained by complex interpolation (with m_{j} and b_{I} varying analytically with respect to the interpolation parameter) between (10) (for a slightly larger value of p) and the estimate

∥ ∑_{I} 2^{−δ(1)(i+s)}m_{i+s}(D)b_{I}∥_{2} ≲ 2^{− \frac{n+1}{2} s} ∑_{I} |I|

proved by Christ [10, 9] under the assumption ∥b_{I}∥_{1} ≲ |I|. (See also [25]).

Similarly, when s ≤ 0 the estimate (9) can be obtained by complex interpolation between (10) and the trivial estimate

∥ ∑_{I} 2^{−δ(p)(i+s)}m_{i+s}(D)b_{I}∥_{L^{1}(R^{n}\ ⋃_{I} CI)} ≲ 2^{−N|s|}(∑_{I} |I|)

for any N > 0, obtained from the triangle inequality and the rapid decrease of m_{i+s}(D)b_{I} outside CI.

4. Proof of the main estimate

It remains to prove (8). If we let J range over all dyadic cubes with sidelength greater than 1 and write f_{J} = χ_{J}f_{j}, where 2^{j} is the sidelength of J, then (8) is clearly equivalent to the estimate

∥ ∑_{J} 2^{δ(p)j}m_{j}(D)f_{J}∥_{p} ≲ (∑_{J} ∥f_{J}∥_{p}^{p})^{1/p}.  (11)

The estimate (11) can be sharp when one or more of the f_{J} is supported on a single c-cube. It turns out that when the f_{J} are spread out over a larger set, then one can get some improvement, in the sense that one upgrade the L^{p} norm on the f_{J} to the L^{p_{0}} norm (at the cost of introducing a small factor into the estimate):

Proposition 4.1. Suppose that each f_{J} obeys the above conditions and is additionally supported on a set of measure 2^{t}, where t is independent of J. Then

∥ ∑_{J} 2^{δ(p)j}m_{j}(D)f_{J}∥_{p} ≲ 2^{εt}(∑_{J} ∥f_{J}∥_{p_{0}}^{p})^{1/p}  (12)

holds for all ε > 0 and p < p_{0}.

This proposition is enough to imply (11). To see this, we note from the identity f_{J} = E(f_{J}) and Hölder’s inequality that ∥f_{J}∥_{∞} ≲ ∥f_{J}∥_{p}; here E is as in Lemma 3.3. Thus we may write f_{J} = ∑_{k≥0} f_{J,k}, where f_{J,k} is the portion of f_{J} on the set where |f_{J}(x)| ∼ 2^{−k}∥f_{J}∥_{p}. By Chebyshev’s inequality we see that each f_{J,k} is supported on a set of measure O(2^{kp}), and by Hölder’s inequality we have

∥f_{J,k}∥_{p_{0}} ≲ 2^{−k( \frac{1}{p} − \frac{1}{p_{0}} )}∥f_{J}∥_{p}.

By Proposition 4.1 we see that

∥ ∑_{J} 2^{δ(p)j}m_{j}(D)f_{J,k}∥_{p} ≲ 2^{εkp}(∑_{J}(2^{−k( \frac{1}{p} − \frac{1}{p_{0}} )}∥f_{J}∥_{p})^{p})^{1/p},

and (11) follows by summing in k, if ε is chosen to be sufficiently small.

It remains to prove Proposition 4.1. We shall fix t and not mention explicity any dependence on t of the various quantities studied in the sequel.

We first prove (12) for the medium-size cubes, where 2^{j} ≲ 2^{Ct} for some large C to be chosen later. Because of the exponential growth term 2^{εt} in the estimate it suffices to show this estimate for a single scale j. Now that we are working with a single scale, it will be convenient to trade the good frequency-localization properties of m_{j} for good spatial-localization properties. Since the convolution kernel of m_{j}(D) is rapidly decreasing outside B(0, 2^{j}) it suffices to consider the portion of the kernel inside this ball, since the remaining terms can be treated by similar arguments but with an additional rapidly decreasing factor in the estimate. If one makes this precise with a smooth cutoff function, this reduction essentially amounts to replacing m_{j}(D) by m̃_{j}(D), where m̃_{j} is some weakly spatially-localized type j multiplier. By localization (see [17]) we may restrict our attention to a single cube J. By (6) we have the estimate

∥2^{δ(p)j}m_{j}(D)f_{J}∥_{p_{0}} ≲ 2^{εj}|J|^{−1/p}|J|^{1/p_{0}}∥f_{J}∥_{p_{0}}

for any ε > 0. By Hölder’s inequality this implies

∥2^{δ(p)j}m_{j}(D)f_{J}∥_{p} ≲ 2^{εj}(∑_{J} ∥f_{J}∥_{p_{0}}^{p_{0}})^{1/p_{0}},

which implies (13) since 2^{j} ≲ 2^{Ct}.

The contribution of the large cubes (with 2^{j} ≫ 2^{Ct}) is more difficult. Ideally, since each f_{J} has relatively small support compared with J, we would like to assume that each f_{J} is supported on a ball of radius 2^{Ct}. Since we cannot do this directly, we will content ourselves with the weaker reduction of assuming that f_{J} is supported on a sufficiently sparse union of 2^{Ct}-balls. We borrow the following lemma from [33].

Definition 4.2. A collection {B(x_{i}, R)}_{i=1}^{M} of R-balls is sparse if the centers x_{i} are R^{C}M^{C} seperated.

Lemma 4.3. [33] Suppose E is the union of c-cubes, and N ≥ 1. For each 0 ≤ k ≤ N define the radii R_{k} inductively by

R_{0} = 1, R_{k+1} = R_{k}^{C}|E|^{C}.

Then for each 0 ≤ k ≤ N − 1 there exists a family of sparse collections {B_{k,α}}_{α∈A_{k}} of balls of radius R_{k}, such that the index sets A_{k} have cardinality O(|E|^{1/N}) and

E ⊆ ⋃_{k=0}^{N−1} ⋃_{α∈A_{k}} ⋃_{B∈B_{k,α}} B.

Proof For each x ∈ E, define the stopping time k(x) to be the least positive integer such that

|E ∩ B(x, R_{k(x)})| ≤ |E|^{\frac{k(x)}{N}};

note that 1 ≤ k(x) ≤ N for all x ∈ E. For each 0 ≤ k ≤ N − 1, let E_{k} = {x ∈ E : k(x) = k + 1} and let {x_{k,α}} be a maximal R_{k}-net of E_{k}. From the estimates

|E ∩ B(x_{k,α}, R_{k+1})| ≤ |E|^{\frac{k+1}{N}} , |E ∩ B(x_{k,α}, R_{k})| ≳ |E|^{\frac{k}{N}} ,

we see that for a fixed k, there can be at most O(|E|^{1/N}) points x_{k,α} in any given ball of radius R_{k+1}. Thus one can cover E_{k} by O(|E|^{1/N}) collections B_{k,α} of R^{k+1}- seperated R_{k}-balls; since one can make the cardinality of these collections O(|E|) these collections are sparse. Since E is covered by ⋃_{k=0}^{N−1} E_{k} this finishes the proof.

Let N be a large number to be chosen shortly. We apply the above lemma for each J with E = E_{J} equal to the support of f_{J}. Thus each E_{J} can be covered by O(N(2^{t})^{1/N}) sparse collections of balls, each of which is associated with a radius R_{k}, 0 ≤ k ≤ N − 1.

Fix the ε appearing in (12), and set N = 1/ε. Since the number of sparse collections is now O_{ε}(2^{εt}), we can prove (12) on each sparse collection seperately (without the 2^{εt} factor) and sum using the triangle inequality. Thus, we may assume that there exists a fixed 0 ≤ k ≤ N such that the support of each f_{J} is covered by a sparse collection B_{J} of R_{k}-balls.

Note that R_{k} ≲ 2^{Ct} for some large C depending on N. In particular, R_{k} ≪ 2^{εj} for some arbitrarily small ε > 0. We henceforth fix k and drop the k-subscripts in the quantities under consideration.

Let ϕ_{B} = ϕ((x − x_{B})/R), where x_{B} is the center of B and ϕ is a radial real Schwarz function which is positive on the unit ball and whose Fourier transform is compactly supported. Then we have f_{J} = ∑_{B∈B_{J}} ϕ_{B}f_{J,B}, where f_{J,B} = χ_{B}f_{J}/ϕ_{B}. Since ϕ_{B} is bounded away from zero on B, we have

∥f_{J}∥_{p_{0}} ∼ ( ∑_{B∈B_{J}} ∥f_{J,B}∥_{p_{0}}^{p_{0}})^{1/p_{0}}

and to prove (12) it will suffice to verify that

∥ ∑_{J} ∑_{B∈B_{J}} 2^{−jδ(p)}m_{j}(D)ϕ_{B}f_{J,B}∥_{p} ≲ (∑_{J}( ∑_{B∈B_{J}} ∥f_{J,B}∥_{p_{0}}^{p_{0}})^{p/p_{0}})^{1/p}.  (13)

To prove (13) we have to exploit some sort of quasi-orthogonality of the operators m_{j}(D)ϕ_{B}; to do this we will factor these operators as the composition of a Fourier restriction operator, an averaging operator, and a Fourier extension operator. It is at this stage that one needs a restriction theorem (4) for the exponent p_{0}. (The extension operator is relatively easy to deal with, due to the averaging effect of the ϕ_{B} and the sparseness of the balls B, and one can get good estimates by standard Tomas-Stein theory).

From the support properties of ϕ̂_{B} and m_{j} and Plancherel’s theorem we have the identity

m_{j}(D)ϕ_{B}f_{J,B} = F^{−1}[m_{j}(ϕ̂_{B} ∗ f̂_{J,B})] = F^{−1}[m_{j}(ϕ̂_{B} ∗ F_{J,B})],  (14)

where F_{J,B} = χ_{A}f̂_{J,B}, A is the annulus {ξ : |1 − |ξ|| ≲ R^{−1}}, and F^{−1} denotes the inverse Fourier transform.

To estimate F_{J,B} we use the restriction theorem (4). By averaging (4) over dilations by 1 + O(R^{−1}) and applying the resulting estimate to f_{J,B} one obtains

∥F_{J,B}∥_{p_{0}} = ∥χ_{A}f̂_{J,B}∥_{L^{p_{0}} (A)} ≲ R^{−1/p_{0}}∥f_{J,B}∥_{p_{0}}.

Comparing this estimate and (14) with (13), one sees that (13) will be proven if one can show that

∥ ∑_{J} ∑_{B∈B_{J}} 2^{−j( \frac{n}{p} − \frac{n+1}{2} )}F^{−1}[m_{j}(ϕ̂_{B} ∗ F_{J,B})]∥_{p} ≲ R^{1/p_{0}}(∑_{J}( ∑_{B∈B_{J}} ∥F_{J,B}∥_{p_{0}}^{p_{0}})^{p/p_{0}})^{1/p}. (15)

We will abstract (15) and prove it for arbitrary functions F_{J,B} and arbitrary 1 ≤ p ≤ p_{0} ≤ 2. By complex interpolation (with m_{j} depending analytically on the interpolation parameter) it suffices to prove (15) for the endpoints (p, p_{0}) = (1, 1), (1, 2), (2, 2). In other words, we show

∥ ∑_{J} ∑_{B∈B_{J}} 2^{−j \frac{n−1}{2}} F^{−1}[m_{j}(ϕ̂_{B} ∗ F_{J,B})]∥_{1} ≲ R ∑_{J} ∑_{B∈B_{J}} ∥F_{J,B}∥_{1};  (16)

∥ ∑_{J} ∑_{B∈B_{J}} 2^{−j \frac{n−1}{2}} F^{−1}[m_{j}(ϕ̂_{B} ∗ F_{J,B})]∥_{1} ≲ R^{1/2} ∑_{J}( ∑_{B∈B_{J}} ∥F_{J,B}∥_{2}^{2})^{1/2}; (17)

∥ ∑_{J} ∑_{B∈B_{J}} 2^{j/2}F^{−1}[m_{j}(ϕ̂_{B} ∗ F_{J,B})]∥_{2} ≲ R^{1/2}(∑_{J} ∑_{B∈B_{J}} ∥F_{J,B}∥_{2}^{2})^{1/2}. (18)

By the triangle inequality (16) reduces to

∥F^{−1}[m_{j}(ϕ̂_{B} ∗ F_{J,B})]∥_{1} ≲ 2^{j \frac{n−1}{2}} R∥F_{J,B}∥_{1}.

By translation invariance we may assume that B is centered at the origin. By Minkowski’s integral inequality it suffices to verify this estimate when F_{J,B} is a delta function δ(· − ξ_{0}). The worst case occurs when ξ_{0} is within O(\frac{1}{R}) of the unit sphere; for simplicity we take ξ_{0} = e_{n} + O(\frac{1}{R}), as the general case can be handled similarly. In this case m_{j}(ϕ̂_{B} ∗ F_{J,B}) becomes a bump function of height O(R^{n}) adapted to a cap on the unit sphere centered at e_{n} with radius R^{−1} and thickness 2^{−j}. A stationary phase computation, combined with the hypothesis 2^{j} ≫ R^{C}, then shows that F^{−1}[m_{j}(ϕ̂_{B} ∗ F_{J,B})] has height O(R^{n}2^{−j \frac{n+1}{2}} ) on the tube \{x : |x| ≲ 2^{j}, |\underline{x}| ≲ R^{−1}2^{j}\}, and is rapidly decreasing away from this tube. Thus the L^{1} norm of this quantity is 2^{j \frac{n−1}{2}} R as desired.

To prove (17) and (18) it suffices to consider a single scale j. For (17) this is possible thanks to the triangle inequality; for (18) one obtains this reduction from the orthogonality estimate

∥ ∑_{j} F^{−1}[m_{j}F_{j}]∥_{2} ≲ (∑_{j} ∥F^{−1}[m_{j}F_{j}]∥_{2}^{2})^{1/2},

which follows from Plancherel’s theorem and the almost disjoint support of the frequency-localized multipliers m_{j}.

Having reduced matters to a single scale, we no longer need the good frequencylocalization properties of the m_{j} and ϕ_{B}, and will trade them in for good spatiallocalization properties. By repeating the argument used previously in this section, we may replace m_{j} by a weakly spatially-localized type j multiplier m̃_{j}. Similarly, one can localize ϕ in physical space, replacing it by a bump function ϕ̃ which is compactly supported in physical space (but no longer compactly supported in frequency space, of course). Thus our estimates (17) and (18) reduce to

∥ ∑_{l(J)=2^{j}} ∑_{B∈B_{J}} F^{−1}[m̃_{j}( ˆϕ̃_{B} ∗ F_{J,B})]∥_{1} ≲ 2^{j \frac{n−1}{2}} R^{1/2} ∑_{l(J)=2^{j}}( ∑_{B∈B_{J}} ∥F_{J,B}∥_{2}^{2})^{1/2};(19)

∥ ∑_{l(J)=2^{j}} ∑_{B∈B_{J}} F^{−1}[m̃_{j}( ˆϕ̃_{B} ∗ F_{J,B})]∥_{2} ≲ 2^{−j/2}R^{1/2}( ∑_{l(J)=2^{j}} ∑_{B∈B_{J}} ∥F_{J,B}∥_{2}^{2})^{1/2}.(20)

However, by Plancherel’s theorem the functions F^{−1}[m̃_{j}( ˆϕ̃_{B} ∗ F_{J,B})] are compactly supported in CJ and thus have almost disjoint support. Therefore, we may localize to a single cube J. But now (19) follows from (20) by Hölder’s inequality. Thus it remains only to prove the estimate

∥ ∑_{B∈B_{J}} F^{−1}[m̃_{j}( ˆϕ̃_{B} ∗ F_{J,B})]∥_{2} ≲ 2^{−j/2}R^{1/2}( ∑_{B∈B_{J}} ∥F_{J,B}∥_{2}^{2})^{1/2}.

Fix J. We will prove this estimate by standard Tomas-Stein theory, taking advantage of the sparseness of the balls B. By Plancherel’s theorem the above estimate is equivalent to

∥ ∑_{B∈B_{J}} m̃_{j}(D)ϕ̃_{B}G_{J,B}∥_{2} ≲ 2^{−j/2}R^{1/2}( ∑_{B∈B_{J}} ∥G_{J,B}∥_{2}^{2})^{1/2}.  (21)

By the T T^{∗} method this estimate is equivalent to

( ∑_{B′∈B_{J}} ∥\overline{ϕ̃_{B′}}|m̃_{j}|^{2}(D)ϕ̃_{B}G_{J,B}∥_{2}^{2})^{1/2} ≲ 2^{−j}R( ∑_{B∈B_{J}} ∥G_{J,B}∥_{2}^{2})^{1/2}.

By Schur’s test and self-adjointness this will obtain if we can show the estimates

∥\overline{ϕ̃_{B}}|m̃_{j}|^{2}(D)ϕ̃_{B}∥ ≲ 2^{−j}R
∥\overline{ϕ̃_{B′}}|m̃_{j}|^{2}(D)ϕ̃_{B}∥ ≲ 2^{−j}R^{−C}M^{−C} when B ≠ B′

if C is sufficiently large; here we use M to denote the cardinality of B_{J}, and ∥ ∥ to denote the L^{2} operator norm.

To prove the former estimate we note that we can localize the kernel of the operator |m̃_{j}|^{2}(D) to the scale R, and replace |m̃_{j}|^{2} by |m̃_{j}|^{2} ∗ ϕ̂_{R}, where ϕ_{R} is a bump function adapted to the ball of radius O(R) around the origin. However a routine computation shows that ∥|m̃_{j}|^{2} ∗ϕ̂_{R}∥_{∞} ≲ 2^{−j}R, and so the desired estimate follows from Plancherel’s theorem.

To prove the latter estimate, we use stationary phase estimates on the convolution kernel of |m̃_{j}|^{2}(D) to obtain the pointwise estimate

|\overline{ϕ̃_{B′}}|m̃_{j}|^{2}(D)ϕ̃_{B}f(x)| ≲ |\overline{ϕ̃_{B′}(x)}| ∫ 2^{−j}|x − y|^{−(n−1)/2} |ϕ̃_{B}(y)| f(y) dy.

The estimate then follows from Schur’s test and the fact that the balls B, B′ are R^{C}M^{C}-seperated.

Remarks.

(i) As observed in [25], the strong-type estimate (8) can show strong-type estimates as well as weak-type estimates. More precisely, we have

Corollary 4.4. For each j, let c_{j} be a complex number and m_{j} be a frequencylocalized type j multiplier. and suppose the Bochner Riesz conjecture holds for some 1 < p_{0} ≤ 2n/(n + 1). Then for any 1 ≤ p < p_{0} and p ≤ q ≤ ∞, the operator ∑_{j} c_{j}m_{j}(D) maps L^{p} to the Lorentz space L^{p,q} if c_{j} ∈ l^{q}.

For q = p this follows immediately from (8), and when q = ∞ one uses a modification of the argument above. The intermediate cases are obtained by real interpolation. This corollary is essentially sharp: translation invariance considerations show that a non-zero multiplier cannot map L^{p} to L^{p,q} if q < p, and the usual bump function example shows that ∑_{j} c_{j}m_{j}(D) does not map L^{p} to L^{p,q} if c_{j} is not in l^{q} and m_{j} is a suitable frequency-localized type j multiplier.

(ii) A careful accounting of epsilons allows the more quantitative version of Theorem 3.2: If 1 < p_{0} ≤ 2n/(n + 1), 0 < ε ≪ 1, and S^{δ(p_{0})+ε} is of strong-type (p_{0}, p_{0}), then S^{δ(p)} is of weak-type (p, p) for all 1 ≤ p ≤ p_{0} − C/ log(1/ε). The presence of the logarithm is caused by the sparse Calderón-Zygmund decomposition, and seems to be an essential feature of this argument.

(iii) Another consequence of (8) is an epsilon-removal result for the Hörmandertype estimate (7) (or the essentially equivalent estimate (6)). More precisely, we have

Corollary 4.5. If (7) holds for some 1 < p_{0} ≤ 2n/(n + 1) and all ε > 0, then (7) also holds for all 1 ≤ p < p_{0} with ε = 0.

The precise relationship between p and ε is as in the previous remark. This result is easily obtained by specializing (8) to a single scale j and rescaling.

If one reviews the proof of (8) for a single scale j, one sees that the technical manipulations involving the strong-j/weak-j distinction and strong orthogonality properties on the Fourier transform side are no longer necessary. In fact the epsilon-removal phenomenon is quite general, and one can replace the phase |x − y| in (7) by any other smooth real function ϕ(x, y) on a bounded domain whose Hessian has constant rank n − 1; analogues for more general phases also exist. We hope to present these results in a later paper.

The author does not know at present whether this epsilon-removal phenomenon also works for (L^{p}, L^{q}) estimates, especially on the sharp line q = \frac{n−1}{n+1}p′. In particular, it seems reasonable that the full Bochner-Riesz conjecture would imply the estimate

∥ ∫ e^{−2πiR|x−y|}a(x, y)f(y) dy∥_{q} ≲ R^{−1/p′}∥f∥_{p}

for all q = \frac{n−1}{n+1}p′ and 1 ≤ p < 2n/(n + 1). By interpolation and the above remarks this estimate would be true for any slightly smaller value of q, and the Knapp example shows that it is false for any larger value of q. (For two dimensions this estimate is proven implicitly in [6]).

(iv) By a slight modification of the above argument, one can “weaken” the restriction theorem hypothesis in Theorem 3.2 to a localized restriction estimate, viz.

∥f̂|_{S^{n−1}}∥_{p_{0}} ≲ R^{ε}∥f∥_{p_{0}}

for all f supported in B(0, R). This weaker estimate is of course also implied by the Bochner-Riesz conjecture at p_{0}, by Theorem 3.1, but this implication can be proved more directly by a simple linearization argument (see [33]).

(v) The statement of Theorem 3.2 can be generalized without modification to handle Bochner-Riesz multipliers that are singular on surfaces other than the sphere, assuming (for instance) that the surface has nonvanishing Gaussian curvature. In particular, the parabolic Bochner-Riesz conjecture is equivalent to its weak-type counterpart, since the parabolic Bochner-Riesz conjecture implies the parabolic restriction conjecture (see [33]). It should be possible to extend this result to general Riesz means of pseudo-differential operators (cf. [26], [12], [24], [32]), although the author has not attempted to do this.

(vi) One can improve the weak-type estimate proven in the above Theorem to a weak weighted inequality of the form

∫_{M_{p}f(x)≥α} |S^{δ(p)}f(x)|^{q} dx ≲ α^{q−p}∥f∥_{p}^{p}

for all 1 ≤ p < p_{0} and some q > p, where M_{p}f = [M(|f|^{p})]^{1/p} and M is the Hardy-Littlewood maximal function; this follows from a modification of the argument used to prove (9). It seems reasonable to conjecture that this estimate holds for all 1 ≤ p < 2n/(n + 1) and p < q ≤ \frac{n−1}{n+1}p′, with the exception of the point p = 1, q = ∞ (or p = 1, q = BMO). In [10] this conjecture was proven for p = 1, q = 2, but by interpolation this can be extended to p = 1, 1 < q < ∞. In [9] and [32] this estimate was shown for q = 2 and 1 ≤ p ≤ 2(n + 1)/(n + 3).

5. On the almost-everywhere convergence of Bochner-Riesz means for p < 2

Fix n ≥ 2. In the previous sections we studied the conjecture that the Bochner-Riesz means S̃^{δ} were bounded on L^{p} for δ > δ(p). By dilation invariance, this is equivalent to the boundedness of S̃_{t}^{δ} on L^{p} uniformly in t > 0 for the same range of p, δ. Since S̃_{t}^{δ}f converges pointwise to f as t → ∞ for test functions f, it is natural to conjecture that

lim_{t→∞} S̃_{t}^{δ}f(x) = f(x) a.e.

for all f ∈ L^{p} and δ > δ(p).

In higher dimensions this conjecture has been verified for p ≥ 2 in [3] using weighted L^{2} spaces, while for p < 2 the conjecture is true for a lacunary choice of t when n = 2, or for radial f for general n (see [3], [18], [19]). However, in general the conjecture fails for p < 2, as we shall now demonstrate.

The conjecture (and various endpoint versions of it) are well-studied in the case p ≤ 1; see e.g. [20], [7]. The conjecture as stated is true for p = 2, but the endpoint case p = 2, δ = 0 remains open and is likely to be extremely difficult, being a higher-dimensional version of the famous Carleson-Hunt theorem [5]. We shall only concern ourselves with the case 1 < p < 2.

By Stein’s maximal principle [28], the pointwise convergence result is equivalent for p ≤ 2 to the maximal Bochner-Riesz estimate

∥ sup_{t>0} S̃_{t}^{δ}f∥_{p,∞} ≲ ∥f∥_{p}.  (22)

We may replace S̃_{t}^{δ} by S_{t}^{δ} = m^{δ}(D/t) as in the previous sections, since the error is pointwise bounded by the Hardy-Littlewood maximal function.

Proposition 5.1. (22) fails when

δ < \frac{2n − 1}{2p} − \frac{n}{2}.  (23)

In particular, the Bochner-Riesz means fail to converge when δ = δ(p) + ε for 1 < p < (2n − 1)/n and sufficiently small ε (depending on p), even when n = 2. This provides a negative answer to Problem 4(a) in [29]. Proof Let R ≫ 1 be a large number and define

f(y) = e^{2πiy_{n}}ψ(\underline{y}, R^{−1/2}y_{n})

where y = (\underline{y}, y_{n}) ∈ R^{n−1} × R and ψ is a non-negative bump function to be chosen later.

Let x = (\underline{x}, x_{n}) be such that 0 < x_{n} ∼ |x| ∼ R. We claim that

|S_{t}^{δ}f(x)| ∼ R^{−δ− \frac{n}{2}} when t = |x|/x_{n}  (24)

if the support of ψ is sufficiently small. This implies that

∥S_{t}^{δ}f∥_{p,∞} ≥ R^{n/p}R^{−δ− \frac{n}{2}} ,

and the result follows by combining this and the estimate ∥f∥_{p} ∼ R^{−1/2p} with (22). To show (24), we write

S_{t}^{δ}f(x) = ∫ K(x − y)e^{2πiy_{n}}ψ(\underline{y}, R^{−1/2}y_{n}) dy,

where standard stationary phase asymptotics give

K(x − y) = c_{+}(x)e^{2πi \frac{|x|}{xn} |x−y|}R^{− \frac{n+1}{2} −δ} + c_{−}(x)e^{−2πi \frac{|x|}{xn} |x−y|}R^{− \frac{n+1}{2} −δ} + o(R^{− \frac{n+1}{2} −δ}),

where |c_{±}(x)| ∼ 1. The contribution of the error term is o(R^{−δ− \frac{n}{2}} ) and can be discarded. The contribution of the second term is rapidly decreasing in R as the combined phase in the integral is non-stationary. The contribution of the first term is

c_{+}(x)R^{− \frac{n+1}{2} −δ} ∫ e^{2πi( \frac{|x|}{xn} |x−y|+y_{n})}ψ(\underline{y}, R^{−1/2}y_{n}) dy.

However if the support of ψ is sufficiently small a Taylor expansion in y shows that \frac{|x|}{x_{n}}|x − y| + y_{n} is within 1/100 of |x|^{2}/x_{n}. Thus there is essentially no phase cancellation and the size of the first term is comparable to R^{−δ− \frac{n}{2}} , as desired.

Based on this proposition it is natural to conjecture

Conjecture 5.2. When 1 < p < 2 (22) holds if and only if δ > 0 and δ ≥ \frac{2n−1}{2p} − \frac{n}{2}.

This conjecture is true for δ > (n − 1)(\frac{1}{p} − \frac{1}{2}) (see [31]) and the critical case is within epsilon of p = (2n − 1)/n, δ = 0. Surprisingly little is known for the remaining cases, even when n = 2 and p = 4/3.

We finish this section by showing that one can localize (22) in space and frequency and reduce matters to the Hörmander-type estimate

∥ sup_{t∼1} ∫ e^{2πiRt|x−y|}a(x, y, t)f(y) dy∥_{p} ≲ R^{δ− \frac{n−1}{2}} ∥f∥_{p},  (25)

for all R ≫ 1, provided one gives up an epsilon in the δ index; here a(x, y, t) is smooth and supported on t ∼ 1, |x−y| ∼ 1, |x|, |y| ≲ 1. Incidentally, a modification of this argument shows that the convergence of a Bochner-Riesz multiplier implies the a.e. convergence of a lacunary sequence of Bochner-Riesz multipliers, with a loss of epsilon in the δ index. The proof has a similar flavor to the sieve argument of Cordoba [14].

Proposition 5.3. Fix 1 < p < ∞. If (22) holds for some δ_{0}, then (25) holds for all δ > δ_{0}. Conversely, if (25) holds for some δ_{0}, then (22) holds for all δ > δ_{0}.

Proof When p ≥ 2 the result follows from a standard application of Littlewood-Paley theory (see e.g. [8]), so we restrict ourselves to the case p < 2.

We first observe that to prove (25) for all a(x, y, t) it suffices to show it for one such amplitude a(x, y, t), for the same reason used to justify the corresponding observation for (7).

To show the former implication, note that by interpolation with known estimates one can replace the weak-type estimate (22) with the corresponding strong-type estimate, giving up an epsilon in the δ index. We may trivially restrict the frequency t in (22) to the range t ∼ 1, and restrict the spatial variables x, y to sets so that |x − y| ∼ R, |x|, |y| ≲ R for some arbitrary R ≫ 1. The estimate (25) now follows by using the asymptotic expansion of the Bochner-Riesz kernel, discarding lower order terms (by considering higher values of δ, or by using Lemma 2.1 in [33]), and rescaling; we omit the details.

Now suppose that (25) holds for some δ_{0}. From (3) we have

S_{t}^{δ} = ϕ(D/t) + ∑_{j>0} 2^{−jδ}m_{j}(D/t)

where m_{j} is a strongly spatially-localized type j multiplier. The contribution of ϕ(D/t) is dominated by the Hardy-Littlewood maximal function and can be discarded. Since L^{p,∞} is a Banach space for p > 1 (see e.g. [31]), to prove (22) it suffices by the triangle inequality to show that

∥ sup_{t} |m_{j}(D/t)f|∥_{p,∞} ≲ j2^{jδ_{0}}∥f∥_{p}

for any j. Henceforth j will be a fixed positive integer.

Let Ω_{k} denote the dyadic interval (2^{k−1}, 2^{k}]. It will be convenient to refine the family of intervals {Ω_{k}} to 2j sub-families {Ω_{2jk+l}}, 0 ≤ l < 2j. By the triangle inequality it suffices to show that

∥ sup ; t∈⋃_{k} Ω_{2jk+l} |2^{−jδ_{0}}m_{j}(D/t)f|∥_{p,∞} ≲ ∥f∥_{p}

for all l. By dilation invariance we may take l = 0.

This estimate is known for p = 2 by the square-function methods discussed in the next section. Thus, by the same Calderón-Zygmund techniques used in Section 2, it suffices to show the estimate

|{ sup ; t∈⋃_{k} Ω_{2jk} | ∑_{I} 2^{−jδ_{0}}m_{j}(D/t)b_{I}| ≳ 1}| ≲ ∑_{I} |I|

for all collections of disjoint dyadic cubes I and all functions b_{I} supported on I such that ∥b_{I}∥_{p} ≲ |I|^{1/p}. By inspection of the Calderón-Zygmund decomposition method one sees that one can impose arbitrary moment conditions on the b_{I}; in particular, we will assume that each b_{I} is orthogonal to any polynomial of degree at most N, where N is a large number to be chosen later.

Let 2^{i} denote the sidelength of I. When i ≥ 2jk+j the expression 2^{−jδ_{0}}m_{j}(D/t)b_{I} is supported on ⋃_{I} CI and so can be discarded:

|{sup_{k} | ∑_{i≥2jk+j} ∑_{l(I)=2^{i}} m_{j}(D/t)b_{I}(x)| ≳ 1}| ≲ ∑_{I} |I|.

When i < 2jk − j the convolution kernel of m_{j}(D/t) is essentially constant at the scale of 2^{i}, and one gets a large gain. More precisely, if one integrates by parts (or performs a Taylor expansion of the kernel of m_{j}(D/t)) and uses the moment conditions one obtains the pointwise estimate

|2^{−jδ_{0}}m_{j}(D/t)b_{I}(x)| ≲ 2^{−(2kj−i)N}|I|χ_{B(x_{I},C2^{2jk})}(x)

for some large N, where x_{I} denotes the center of I. Summing this over all I of a fixed sidelength and assuming N is large enough to absorb all other factors, one obtains

| ∑_{l(I)=2^{i}} m_{j}(D/t)b_{I}(x)| ≲ 2^{−(2kj−i) \frac{N}{2}}

for all i < 2jk − j. Summing this over all such i and taking the supremum over k, one obtains

sup_{k} | ∑_{i<2jk−j} ∑_{l(I)=2^{i}} m_{j}(D/t)b_{I}(x)| ≲ 2^{−jN/2},

and so the contribution of these terms is acceptable.

To finish the proof we have to deal with the terms when 2jk − j ≤ i < 2jk + j. In other words, we need to show

|{sup_{k} | ∑_{2k−j≤i<2jk+j} ∑_{l(I)=2^{i}} m_{j}(D/t)b_{I}(x)| ≳ 1}| ≲ ∑_{I} |I|.

For each I let k(I) be the unique integer such that 2jk − j ≤ i < 2jk + j. We rewrite the above estimate as

|{sup_{k} sup_{t∈Ω_{2jk}} | ∑_{I:k(I)=k} 2^{−jδ_{0}}m_{j}(D/t)b_{I}| ≳ 1}| ≲ ∑_{k} ∑_{I:k(I)=k} |I|.

It suffices to prove this for a single k:

|{ sup_{t∈Ω_{2jk}} | ∑_{I:k(I)=k} 2^{−jδ_{0}}m_{j}(D/t)b_{I}| ≳ 1}| ≲ ∑_{I:k(I)=k} |I|.

By dilation invariance we may take k = 0. By Chebyshev’s inequality it suffices to show that

∥ sup_{t∼1} |2^{−jδ_{0}}m_{j}(D/t) ∑_{I:k(I)=0} b_{I}|∥_{p} ≲ ∥ ∑_{I:k(I)=0} b_{I}∥_{p}.

Let f = ∑_{I:k(I)=0} b_{I}. Since m_{j}(D/t) is local at the scale of 2^{j} we can localize this (L^{p}, L^{p}) estimate (cf. [17]) and reduce our estimate to

∥ sup_{t∼1} |2^{−jδ_{0}}m_{j}(D/t)f|∥_{L^{p}(B(0,C2^{j}))} ≲ ∥f∥_{p}

for all f supported on B(0, C^{−1}2^{j}). The estimate now follows by rescaling (25) for R = 2^{j}, and noting that the convolution kernel of m_{j}(D/t) is of the form 2^{−j(n+1)/2}e^{±2πit|x|}a_{±}(t, 2^{−j}|x|), modulo higher order terms which can be easily dealt with (cf. the argument for the other implication in this proposition).

6. A linearization of the local smoothing conjecture

For all test functions f, g, let u denote the solution to the homogenous Cauchy problem for the wave equation

\{ \begin{matrix}(−\frac{∂^{2}}{∂t^{2}} + Δ)u(t, x) = F (t, x), & (t, x) ∈ R × R^{n} \\ u(0, ·) = f, ∂_{t}u(0, ·) = g; \end{matrix}

equivalently, one may write

u(t) = cos(2πt|D|)f + \frac{sin(2πt|D|)}{|D|}g.  (26)

The local smoothing conjecture of Sogge [27] states that

∥u∥_{L^{p}([1/2,2]×R^{n})} ≲ ∥f∥_{L_{δ}^{p}} + ∥g∥_{L_{δ−1}^{p}}  (27)

for all δ > δ(p) and p ≥ 2, where L_{α}^{p} is the Sobolev space ⟨D⟩^{−α}L^{p}, with ⟨ξ⟩ = (1 + |ξ|^{2})^{1/2}. (For p < 2 there is no local smoothing, and (27) is only true when δ > (n − 1)(\frac{1}{p} − \frac{1}{2})).

This conjecture is related to many estimates for the wave equation and the spherical maximal function (see e.g. [23]), but we shall concentrate on the relationship between this conjecture and the Bochner-Riesz family of conjectures. We sketch the argument in Proposition 7.3 of [22]. By Hölder’s inequality (27) implies

∥(∫_{t∼1} |u(t, ·)|^{2} \frac{dt}{t})^{1/2}∥_{L^{p}(R^{n})} ≲ ∥f∥_{L_{δ}^{p}} + ∥g∥_{L_{δ−1}^{p}},

By (26) this is equivalent to

∥(∫_{|t|∼1} |e^{2πit|D|}f|^{2} \frac{dt}{t})^{1/2}∥_{L^{p}(R^{n})} ≲ ∥⟨D⟩^{δ}f∥_{p}.  (28)

We may rescale this as

∥(∫_{|t|∼T} |e^{2πit|D|}f|^{2} \frac{dt}{t})^{1/2}∥_{L^{p}(R^{n})} ≲ ∥⟨T D⟩^{δ}f∥_{p}

for any T > 0.

Now assume temporarily that the Fourier transform of f is supported in the annulus |ξ| ∼ 2^{j} for some j, so that ∥⟨T D⟩^{δ}f∥_{p} ∼ (1 + 2^{j}T)^{δ}∥f∥_{p}. By summing the above estimate over all dyadic T we obtain

∥(∫ \frac{|2^{j}t|}{(1 + |2^{j}t|^{2})^{α}} |e^{2πit|D|}f|^{2} \frac{dt}{t})^{1/2}∥_{L^{p}(R^{n})} ≲ ∥f∥_{p}

for any α > δ + \frac{1}{2}.

By Plancherel’s theorem and Cauchy-Schwarz we have the pointwise estimate

|m(D/τ)f(x)| ≲ (∫ \frac{|2^{j}t|}{(1 + |2^{j}t|^{2})^{α}} |e^{2πit|D|}f(x)|^{2} \frac{dt}{t})^{1/2}

for any radial multiplier m in the Sobolev space L_{α}^{2}([1/2, 2]), where τ ∼ 2^{j} is arbitrary. Since m̃^{δ+ε} ∈ L_{α}^{2}([1/2, 2]) for all ε > 0 and suitable α, we thus have that

∥ sup_{τ∼2^{j}} |S_{τ}^{δ+ε}f|∥_{p} ≲ ∥f∥_{p}.

We may remove the restriction on τ and f by Littlewood-Paley theory, since p ≥ 2. Thus any local smoothing estimate for p ≥ 2 implies the corresponding estimate for the maximal Bochner-Riesz operator except for an epsilon loss in the δ exponent. In particular, the full local smoothing conjecture implies the full Bochner-Riesz conjecture, since the Bochner-Riesz multiplier is self-adjoint.

In this section, we use a linearization method to show that the local smoothing conjecture also implies a restriction theorem for the hyperplane Radon transform. More precisely, we have an estimate of the form

∥RF ∥_{L^{p′} ([−1,1]×S^{n−1})} ≲ ∥F ∥_{L_{α}^{p′} (R×R^{n})}  (29)

for all F supported in a fixed compact subset of R × R^{n}, where R is the operator

RF (s, ω) = ∫_{R_{n}} F (s − ⟨y, ω⟩, y) dy  (30)

for s ∈ R, ω ∈ S^{n−1}. As we shall shortly see, the local smoothing conjecture implies (29) for all p ≥ 2n/(n − 1) and all α > −n/p.

The operator R, when restricted to distributions F on the boundary of an obstacle, appears naturally in the scattering theory for the wave equation; see [21].

A similar linearization was used in [33] to show that the Bochner-Riesz conjecture implied the restriction conjecture. To close this circle of ideas we will also show that the Radon transform conjecture implies the restriction conjecture.

Theorem 6.1. Let 2n/(n − 1) ≤ p < ∞. If (27) is true for all δ > δ(p), then then (29) holds for all α > −n/p.

Theorem 6.2. Let 2n/(n − 1) ≤ p_{0} < ∞. If (29) holds for p = p_{0} and all α > −n/p_{0}, then (4) holds for all 1 ≤ p < p′_{0}.

Proof (Of Theorem 6.1) By (26) the hypothesis (27) is equivalent to

∥e^{2πit|D|}f∥_{L^{p}([1/2,2]×R^{n})} ≲ ∥⟨D⟩^{δ}f∥_{p}.

It will be convenient to recast this estimate in terms of spherical averages. (For a discussion of this connection see [23]). Let A_{t} denote the averaging operator

A_{t}f(x) = ∫_{S^{n−1}} f(x + tω) dω.

From the asymptotics of the Fourier transform of the sphere we see that

A_{t} = (a_{+}(t, x, D)e^{2πit|D|} + a_{−}(t, x, D)e^{−2πit|D|})⟨D⟩^{− \frac{n−1}{2}}

for t ∼ 1, where a_{+}, a_{−} are standard symbols of order 0. Since such symbols are bounded on L^{p} we can rewrite the local smoothing conjecture as

∥A_{t}f∥_{L^{p}([1/2,2]×R^{n})} ≲ ∥⟨D⟩^{δ− \frac{n−1}{2}} f∥_{p}.

By duality this is equivalent to

∥ ∫_{t∼1} A_{t}F (t) dt∥_{p′} ≲ ∥⟨D⟩^{δ− \frac{n−1}{2}} F ∥_{L^{p′} ([1/2,2]×R^{n})}.

Let k ≫ j > 0 be large numbers to be fixed later, and let β(ξ) be a Littlewood-Paley cutoff supported in |ξ| ∼ 1. By localizing to frequency 2^{j+k} we obtain

∥ ∫_{t∼1} A_{t}β(2^{−j−k}D)F (t) dt∥_{p′} ≲ 2^{(j+k)(δ− \frac{n−1}{2} )}∥F ∥_{p′}.

We apply this estimate to a function of the form

F (t, x) = G(2^{k}(t − 1), 2^{k}x),

where G is supported on a fixed compact set. We also localize the left-hand side to the annulus

{(1 + 2^{−k}s)ω : s ∈ [−1, 1], ω ∈ S^{n−1}},

since we expect A_{t}β(2^{−j−k}D)F (t) to be concentrated in this annulus. Reparameterizing the left-hand side in terms of s and ω, and performing some other rescalings, we see that the desired estimate can be written as

2^{−k/p′}∥2^{−k} ∫_{τ∼1} A_{1+2^{−k}τ} β(2^{−j−k}D)F (1 + 2^{−k}τ, (1 + 2^{−k}s)ω) dτ∥_{L^{p′} ([−1,1]×S^{n−1})}
≲ 2^{(j+k)(δ− \frac{n−1}{2} )}2^{− \frac{n+1}{p′} k}∥G∥_{p′}.

Writing δ = δ(p) + ε = \frac{n−1}{2} − \frac{n}{p} + ε and using the identity

A_{1+2^{−k}τ} β(2^{−j−k}D)F (1 + 2^{−k}τ, (1 + 2^{−k}s)ω) = A_{2^{k}+τ} β(2^{−j}D)G(τ, (2^{k} + s)ω)

we can simplify this to

∥ ∫_{τ∼1} 2^{(n−1)k}A_{2^{k}+τ} β(2^{−j}D)G(τ, (2^{k} + s)ω) dτ∥_{p′} ≲ 2^{− \frac{n}{p} j}2^{εj}2^{εk}∥G∥_{p′}.

We now perform the linearization step, approximating the spherical average by a hyperplane average. More precisely, we claim that

2^{(n−1)k}A_{2^{k}+τ} β(2^{−j}D)G(τ, (2^{k} + s)ω)

= ∫ β(2^{−j}D)G(τ, y)δ(⟨y, ω⟩ − s + τ) dy + O(2^{Cj}2^{−k}∥G∥_{p′})

for some C, where δ denotes the Dirac distribution. This estimate reflects the geometric fact that the sphere of radius 2^{k} + τ around (2^{k} + s)ω is within O(2^{−k}) of the hyperplane {y : ⟨y, ω⟩ = s − τ} on the support of G. The error incurred by approximating the sphere by the hyperplane is controlled by 2^{−k} times the derivative of the kernel of β(2^{−j}D), convolved with G, and if one is willing to lose several powers of 2^{j} one can control this uniformly by 2^{−k}∥G∥_{p′}. There is also an error arising from the fact that β(2^{−j}D) does not quite have compact support, but that term is extremely small.

Applying this approximation to our estimate we obtain

∥ ∫_{τ∼1} ∫ β(2^{−j}D)G(τ, y)δ(⟨y, ω⟩−s+τ) dydτ∥_{p′} ≲ 2^{− \frac{n}{p} j}2^{εj}2^{εk}∥G∥_{p′}+O(2^{Cj}2^{−k}∥G∥_{p′}).

By optimizing this estimate in k and ε we obtain

∥Rβ(2^{−j}D)G∥_{L^{p′} ([−1,1]×S^{n−1})} ≲ 2^{− \frac{n}{p} j}2^{εj}∥G∥_{p′}

for all ε, j > 0, where R is the Radon transform (30). For j ≤ 0 one can use crude estimates (i.e. replacing everything by absolute values) to obtain the bound

∥Rβ(2^{−j}D)G∥_{L^{p′} ([−1,1]×S^{n−1})} ≲ 2^{nj}∥G∥_{p′}.

Thus on summing in j and using the triangle inequality one obtains (29) for all α > −\frac{n}{p}, as desired.

Proof (Of Theorem 6.2) We apply (29) to a function of the form

F (t, x) = e^{2πiλt}f(λx)

for some λ ≫ 1, where f is any function supported on B(0, λ). Since

RF (s, ω) = e^{2πiλs} ∫_{R_{n}} e^{−2πiλ⟨y,ω⟩}f(λy) dy = λ^{−n}e^{2πiλs}f̂(ω),

the estimate (29) becomes

∥f̂|_{S^{n−1}}∥_{p′_{0}} ≲ λ^{n}∥F ∥_{L_{α}^{p′_{0}}} .

By replacing f with β(D)f, where β is a Littlewood-Paley cutoff that is equal to 1 on the unit sphere, this estimate can be seen to reduce to

∥f̂|_{S^{n−1}}∥_{p′_{0}} ≲ λ^{n}λ^{α}∥F ∥_{p′_{0}}.

Since α can be made arbitrarily close to −n/p by hypothesis, this reduces to

∥f̂|_{S^{n−1}}∥_{p′_{0}} ≲ λ^{ε}∥f∥_{p′_{0}}

for all ε > 0. To remove the epsilon and obtain the desired conclusion (with a slightly weaker exponent p) one applies Theorem 1.2 of [33], which is an epsilonremoval result proven in a similar fashion to the results in Sections 2 and 3 of this paper.

Remarks.

(i) Both of the above theorems can be extended to more general Fourier integral operators. Suppose that F is a Fourier integral operator of the form

Ff(x) = ∫ e^{2πiϕ(x,ξ)}a(x, ξ)f̂(ξ) dξ,

where x ∈ M and ξ ∈ (R^{n})^{∗}, M is an (n + 1)-dimensional manifold a is a standard elliptic symbol of order δ, and ϕ is a phase function satisfying the cinematic curvature condition (see e.g. [22]). Fix an x_{0} in the support of a, and let Σ ⊂ T_{x_{0}}^{∗}M be the cone {ϕ_{x}(x_{0}, ξ) : ξ ∈ (R^{n})^{∗}}, endowed with some smooth surface measure away from the origin. Suppose that one has an estimate of the form

∥Ff∥_{L^{p}(M)} ≲ ∥f∥_{L^{p}(R^{n})}

where δ = \frac{n−1}{2} − \frac{n}{p} + ε for some small ε. Then for all x_{0} ∈ R^{n+1} we have

∥R_{[x_{0}]}F ∥_{L^{p′} (Σ)} ≲ ∥F ∥_{L_{α}^{p′} (T_{x_{0}} M)}

for all F in a fixed compact subset of T_{x_{0}}M, where

R_{[x_{0}]}F (η) = ψ(η) ∫ F (x)δ(⟨x, η⟩ − 1) dx,

ψ is a bump function supported away from the origin, and α > −n/p + 2ε. Thus every “sharp” Fourier integral operator estimate implies a corresponding restriction theorem for the hyperplane Radon transform for each x_{0}.

Furthermore, if the above Radon transform estimate is true for all ε > 0, then one has a Fourier restriction theorem for any conic section of Σ, More precisely, if S is a hypersurface with boundary in R^{n} which is affinely isomorphic to a compact subset of Σ∩π for some hyperplane π not containing the origin, then one has

∥f̂|_{S}∥_{q} ≲ ∥f∥_{q}

for all 1 ≤ q < p′; the constant of course depends on S.

We hope to present these results more fully in a future paper. (ii) One can similarly linearize the square-function estimate (28). If (28) holds

for some 2n/(n − 1) ≤ p < ∞ and all δ > δ(p), then (29) holds as before, but with the L^{p′} norm of F (t, x) on the right replaced by an L_{x}^{p′}L_{t}^{2} norm. This weaker Radon transform estimate is still sufficient to imply the spherical restriction theorem.

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Department of Mathematics, UCLA, Los Angeles, CA 90024

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