The almost everywhere convergence of

Bochner-Riesz multipliers in the plane

for p < 2

Terence Tao Mathematics Department UCLA

     www.math.ucla.edu/∼tao/preprints

The purpose of this talk is to:

• State the problem of almost-everywhere convergence of Bochner-Riesz multipliers in the plane

• State the classical positive and negative results

• Show some new positive and negative results Classical theory of Fourier summation (one dimension)

• Let f be an L^{p}(R) function (1 < p < ∞), t > 0, and consider the Dirichlet operator

S_{t}f(x) = ∫_{|ξ|<t}e^{2πix·ξ}f̂(ξ) dξ.

• For smooth functions S_{t}f converges to f uniformly as t → ∞. What happens if f is merely in L^{p}?

• S_{t}f converges to f in L^{p}. (Hardy?)

• S_{t}f converges a.e. to f. (Carleson-Hunt)

• Can speed up the rate of convergence by taking a Riesz mean instead of a Dirichlet operator:

S_{t}^{δ}f(x) = ∫ e^{2πix·ξ}(1 − \frac{|ξ|}{t})_{+}^{δ}f̂(ξ) dξ.

When δ = 0 this is the Dirichlet mean; when δ = 1 this is the Fejer mean. The higher the δ, the better the convergence.

What happens in two dimensions?

• Suppose we define the same operator in two dimen-

sions:

S_{t}f(x) = ∫_{|ξ|<t}e^{2πix·ξ}f̂(ξ) dξ.

This is called the disc multiplier.

• When f is smooth, S_{t}f(x) still converges to f uniformly. What happens if f is merely in L^{p}?

• Does S_{t}f converge to f in L^{p} in general? NO unless p = 2! (Fefferman’s example)

• Does S_{t}f converge a.e. to f? NO if p < 2 (Fefferman’s example) UNKNOWN if p = 2. (p > 2 is equivalent to p = 2). This problem is very difficult.

• What about Bochner-Riesz means of order δ > 0?

S_{t}^{δ}f(x) = ∫ e^{2πix·ξ}(1 − \frac{|ξ|}{t})_{+}^{δ}f̂(ξ) dξ.

The “easy” problem: L^{p} convergence of Bochner-Riesz means for δ > 0.

• Under what conditions on p, δ does S_{t}^{δ}f converge to f in L^{p} for all f ∈ L^{p}?

• By some soft analysis, this is equivalent to proving

the estimate

∥S_{1}^{δ}f∥_{p} ≲ ∥f∥_{p}

for test functions f.

• Take f to be a standard bump function. Then a stationary phase (or Bessel function) computation shows that |S_{1}^{δ}f(x)| decays like (1 + |x|)^{−δ−\frac{1}{2}}. Thus a necessary condition is that δ + \frac{1}{2} > \frac{2}{p′}. (Herz)

• By duality we also obtain the condition δ + \frac{1}{2} > \frac{2}{p}. (Herz)

• When δ > 0, these necessary conditions are also suffiicent.

• If δ > 1/2 then the kernel of S_{1}^{δ} turns out to be integrable, and the result follows from Young’s inequality. (Bochner)

• If p = 2 and δ > 0 then the result follows trivially from Plancherel’s theorem.

• When p = 4, δ > 0, the result follows by explicitly computing the L^{4} norm (Carleson, Sjölin, Fefferman, 1972). When p = 4/3 the result follows by duality.

• By interpolation between all of these results, one obtains the sharp sufficient conditions on p and δ.

The “hard” problem: almost everywhere convergence of Bochner-Riesz means for δ > 0

• When p ≤ 2, this problem is equivalent to the maximal function estimate

∥ sup_{t>0} |S_{t}^{δ}f|∥_{p,∞} ≲ ∥f∥_{p}

(Stein, 1961). When p > 2 the equivalence is not perfect, but pretty good (Garcia de Cuerva, Rubio de Francia, ∼ 1980).

• The same necessary conditions δ + \frac{1}{2} > \frac{2}{p}, \frac{2}{p′} can be shown to be true. Are they sufficient? (Stein, 1978)

• When δ > 1/2 the kernels S_{t}^{δ} are integrable, and the result follows for 1 < p < ∞ from the Hardy-Littlewood maximal inequality.

• When p = 2, δ > 0 one can replace the supremum by a square function using Sobolev embedding, and then use Plancherel’s theorem. (Stein, 1961)

• When p = 4, δ > 0, the result follows by replacing the supremum by a square function and explicitly computing the L^{4} norm (Carbery, 1983).

• Duality doesn’t work anymore!

The necessary conditions above are not sufficient when p < 2!

• Stein’s conjecture is true for radial functions (Kanjin, Kojima, 1983) or lacunary approaches (Carbery, 1986), but fails in general!

• Theorem (T., 1997): The estimate

∥ sup_{t∼1}|S_{t}^{δ}f|∥_{p,∞} ≲ ∥f∥_{p}

(and thus almost everywhere convergence) fails when δ < \frac{3}{2p} − 1.

• Consider first the Herz example, in which f is a bump function. The convolution kernel of S_{t}^{δ} is essentially e^{±2πi|x|/t}|x|^{−1/2−δ} when t ∼ 1, and so S_{t}^{δ}f(x) ∼ e^{±2πi|x|}|x|^{−1/2−δ} (there is no cancellation). This gives the Herz condition δ > \frac{2}{p} − \frac{3}{2}.

• To improve upon the Herz example, we’d like to widen the support of f, however one then runs into the cancellation in the kernel. To improve upon this example we have to do something non-radial.

• Let f = e^{ix_{1}}ψ(x_{1}, x_{2}/R) for some large R and a bump function ψ. If there were no cancellation, then S_{t}^{δ} ∗ f(x) would be about R|x|^{−1/2−δ} in size for |x| ≫ R.

• When |x| ∼ R^{2} then the above estimate is sharp for a special value of t (depending on x), thanks to a Taylor series approximation.

What morals can we draw from this counterexample?

• For sake of discussion let’s restrict ourselves to the unit frequency situation t ∼ 1.

• Bad things happen when f has small support (say R) and the piece of the kernel we are interested in has large support (say R^{2}).

• Bad things happen when the support of f is thin (i.e. unit thickness).

• Hopefully: one can show that the estimates improve when the support of f is large and thick.

Positive results for 1 < p < 2

• From the known results when p = 1 or p = 2 and complex interpolation we know that one has a.e. convergence when δ > \frac{1}{p} − \frac{1}{2}. (Stein, 1956). First use of complex interpolation!

• The case p ≤ 1 was handled by Stein, Taibleson, and Weiss in 1981. (The best results use Hardy spaces instead of Lebesgue spaces).

• From the above counterexample one does not have a.e. convergence when δ < \frac{3}{2p} − 1.

• Theorem: (T., 1998) One has a.e. convergence when δ > max(\frac{3}{4p} − \frac{3}{8}, \frac{7}{6p} − \frac{2}{3}).

• We will sketch the proof of this theorem in the following slides.

12

Preliminary reduction

• For the given range of δ, p we have to prove the esti-

mate

∥ sup_{t>0} |S_{t}^{δ}f|∥_{p,∞} ≲ ∥f∥_{p}.

• Note that the frequency t is not localized, and neither is the kernel of S_{t}^{δ}. However, it turns out that one can use a combination of Littlewood-Paley theory and Calderón-Zygmund theory to localize the frequency and kernel respectively, if one is willing to lose an epsilion in the δ index (and that’s not a problem).

• By rescaling we can localize the frequency to t ∼ λ and the kernel to size 1. The estimate then becomes (ignoring epsilons)

∥Sf∥_{L_{x}^{p}L_{t}^{∞}} ≲ λ^{−\frac{1}{2}+δ}∥f∥_{p}

where

Sf(x, t) = ∫ e^{2πiλt|x−y|}a(x, y, t)f(y) dy

and a(x, y, t) is any smooth function supported on the region t, |x − y| ∼ 1, |x|, |y| ≲ 1.

Sf(x, t) = ∫ e^{2πiλt|x−y|}a(x, y, t)f(y) dy

• Trivially we have the estimate

∥Sf∥_{L_{x}^{∞}L_{t}^{∞}} ≲ ∥f∥_{1}.  (1)

This estimate is sharp as long as there is no cancellation in Sf.

• Also, we have

∥Sf∥_{L_{x}^{2}L_{t}^{∞}} ≲ λ^{−\frac{1}{2}}∥f∥_{2}  (2)

This is obtained by using Sobolev embedding to replace the L_{t}^{∞} norm with an L_{t}^{2} norm, then using some L^{2} theory. This estimate is also sharp, as shown by the Knapp example f = e^{iλx_{1}}ψ(x_{1}, λ^{1/2}x_{2}).

• Interpolating between these two estimates gives Stein’s classical result.

• In this rescaled problem the analogue of the improved Herz example is f = e^{iλx_{1}}ψ(λ^{1/2}x_{1}, λx_{2}). This example is sharp for (1), but not sharp for (2) (which is why there is a gap between the negative results from this example, and the Stein estimates). In fact you gain a factor of λ^{−\frac{1}{4}} in (2) for this example.

How to improve upon the classical estimates?

∥Sf∥_{L_{x}^{∞}L_{t}^{∞}} ≲ ∥f∥_{1}.  (1)
∥Sf∥_{L_{x}^{2}L_{t}^{∞}} ≲ λ^{−\frac{1}{2}}∥f∥_{2}  (2)

• By Marcinkeiwicz interpolation we may take f to be a characteristic function f = χ_{E}.

• When E is very small then (1) is superior to (2), but when E is very large then (2) is superior to (1). The only time the two are comparable is when |E| ∼ λ^{−1}; in any other instance, one can do better than just interpolating (1) and (2).

• Because the kernel of S oscillates with wavelength λ^{−1}, the uncertainty principle allows us to assume that E is the union of “dots” of diameter λ^{−1}. By the previous reduction there are about λ of these dots.

• The main difficulty is the sup norm L_{t}^{∞} on the lefthand side. For p ≥ 2 one can use Sobolev embedding and replace it with an L_{t}^{2} norm, but this method is very inefficient for p < 2 (in fact one can show that you can’t do better than Stein’s estimates this way!).

• Since (2) is not sharp for the improved Herz example, the obvious thing to do is to look for an improvement to (2). Based on the Knapp and improved Herz examples, one expects an improvement to (2) whenever the support of f is small. (For instance, if f is supported on a λ^{−1}-ball, then one gets a gain of λ^{−\frac{1}{2}} in (2) just from (1) and Hölder’s inequality.) In fact, we have more:

• Lemma. If f is supported in a small square Q then

∥Sf∥_{L_{x}^{2}L_{t}^{∞}} ≲ λ^{−\frac{1}{2}}|Q|^{1/4}∥f∥_{2}.  (2′)

This is proved by rescaling Q to the unit ball and then applying the argument used to prove (2). (The difference is that Sobolev’s inequality becomes more favourable).

• (2’) is sharp with respect to all known examples, which is a good sign.

• This estimate shows that one can improve upon Stein’s estimate if f is concentrated in a small square, or perhaps a small number of small squares. If E is contained in a collection of disjoint squares Q such that |E ∩ Q| ≳ α|Q|^{1/2}, then (2’) and the triangle inequality give

∥Sχ_{E}∥_{L_{x}^{2}L_{t}^{∞}} ≲ λ^{−\frac{1}{2}}α^{−1}|E|.  (3)

This is a good estimate if α is large, say α ≫ λ^{−1}.

• This motivates the following idea: use a Calderón-Zygmund decomposition to break E up into a collection of sets concentrated in small squares Q, for which |E ∩ Q| ≳ α|Q|^{1/2}, plus a sparsely distributed remainder, for which |E∩Q| ≲ α|Q|^{1/2} for all squares Q.

• If α ≫ λ^{−1} then the concentrated part can be dealt with favourably by (3). Thus we may assume that E is sparse in the sense that

|E ∩ Q| ≲ λ^{−1}|Q|^{1/2}.

Heuristically, this means that E is the λ^{−1}-neighbourhood of a set of dimension at most 1.

• This still gives a lot of possibilities as to the shape of E: it could be clustered around a curve, or in a lattice, or something in between. The two extreme possibilities are given as follows:

• If we can find an argument which improves upon the classical estimates on both of these extreme cases, then we can probably handle the general case.

The lattice case

• If we can show any cancellation whatsoever, then we can improve upon the L^{1} estimate (1), and thus improve upon the Stein estimates.

• Let’s only look at the contribution of E on a single λ^{−1/2} × 1 rectangle, and look at Sχ_{E} on a single λ^{−1/2} square which lies in the double of that rectangle. If there is cancellation for the typical example of this situation, then there is significant cancellation overall.

• The significance of this example is that the phase has been linearized: λt|x − y| essentially becomes λt(x_{1} − y_{1}). The operator is now basically a Fourier transform evaluated at λt.

• However, Plancherel’s theorem states that the number of frequencies in which no cancellation occurs in the Fourier transform is proportional to the support of the function:

|{ξ : |χ̂_{A}(ξ)| ∼ |A|}| ≲ |A|^{−1}.

So, if we wish to avoid cancellation, t must be essentially restricted to a set of measure λ^{−1/2}.

• If t was constant then we’re just studying a single Bochner-Riesz operator, and one gets a huge amount of cancellation from the standard estimates (especially the Carleson-Sjölin L^{4} estimate), partly because E is so sparse. The cancellation is so great that there is still some left if t is only restricted to a set of measure λ^{−1/2} as above. Thus we do get some cancellation, and thus some gain over the Stein estimates.

The thin case

• Now suppose the set E is concentrated as much as possible while still obeying the sparseness condition

|E ∩ Q| ≲ λ^{−1}|Q|^{1/2}.

A good model of this occurs when E is clustered near a one-dimensional set.

• Now one restricts E to a λ^{−1/2} square Q and investigates the possibility of cancellation. Let’s say the square is centered at the origin, and contains a fair amount of E (say about λ^{1/2} dots of diameter λ^{−1/2}). In this case the phase can be approximated by

λt|x − y| = λt|x| − ⟨y, λt\frac{x}{|x|}⟩ + O(1)

Discarding the unimportant λt|x| term, the operator thus becomes a Fourier transform evaluated at λt\frac{x}{|x|}.

• By invoking Plancherel’s theorem as before, we conclude that t is again restricted to a set of measure λ^{−1/2} for “most” directions x/|x|.

• But we have several squares Q with this property. An argument of Córdoba can then be used to show that t is restricted to sets of measure λ^{−1/2} for most x. (This is related to the fact that Besicovitch sets have full dimension in the plane).

• Now one can use the standard L^{4} theory as before.

• Intermediate cases, in which E is neither totally sparse or totally one-dimensional, can be handled by a mix of the above two arguments.