GOING TO EXTREMES

TERENCE TAO

Abstract. We demonstrate some examples of a useful technique in proving an estimate, namely that of passing to an extreme instance of that estimate. This note has much in common with [?].

1. Introduction

This note will be concerned with harmonic analysis applications and variants of the following basic principle:

Extremization principle Let f be a continuous function on a compact set. In order to prove an estimate |f(x)| ≲ M for all x, it suffices to prove this under the assumption that x is at a global maximum of f.

The point of this principle is that the global maximum assumption allows us to deduce additional properties of x, such as f′(x) = 0, which can be helpful in proving the estimate |f(x)| ≲ M.

Technically, we have to verify the continuity and compactness conditions before we can be sure that f actually attains a maximum, which could in principle cause a circular argument. (For instance, if f is a linear operator, to prove continuity we have to prove a bound such as |f(x)| ≲ M on the unit ball to begin with!). However, this problem is usually avoidable either by using a near-maximum (something within ε of the supremum) instead of the maximum, or assuming some sort of a priori bound, or passing to a discretized model in which everything in sight is finite and so maxima are attained.

Here’s one typical application. Suppose T is a self-adjoint operator, and we wish to prove that T is bounded on L^{2}, so that ∥T f∥_{2} ≲ 1 for all f such that ∥f∥_{2} = 1. An easy application of the extremization principle shows that we may assume without loss of generality that f is an eigenvector of T, so that T f = λf. As an application of this reduction, we see that it suffices to bound T^{2}f, T^{100}f in L^{2} in order to obtain the desired estimate. (If T is positive, it even suffices to bound things like e^{T} f).

Here’s the “restricted weak-type” version of the above application. Suppose we wish to prove that a linear operator T is restricted weak-type (p, q) for some 1 < p, q < ∞, so that

|{|T χ_{E}| ∼ α}| ≲ α^{−q}|E|^{q/p}.

For simplicity let us take T to have a non-negative kernel, so that T χ_{E} is nonnegative. By writing the set on the left-hand side as F and doing some algebra, we see that it thus suffices to show that

1991 Mathematics Subject Classification. 42B.

⟨T χ_{E}, χ_{F} ⟩ ≲ |E|^{1/p}|F |^{1/q′}  (1)

under the uniformity assumption T χ_{E} ∼ α on F, for some α, which we can then compute as

α = \frac{⟨T χ_{E}, χ_{F} ⟩}{|F |}.  (2)

By duality, one can also reduce to proving (1) under the dual assumption, namely that T^{∗}χ_{F} ∼ β on E, where

α = \frac{⟨T χ_{E}, χ_{F} ⟩}{|F |}.  (3)

It would be nice if one could somehow have both uniformity assumptions hold simultaneously. I don’t know of a way to achieve this directly, but there are two good substitutes. One is to repeatedly take refinements of E and F; thus we may have T χ_{E} ∼ α on F, T χ_{F} ∼ β on E′ ⊂ E, T χ_{E′} ∼ α on F′ ⊂ F, T χ_{F′} ∼ β on E′′ ⊂ E′, etc. where E, E′, E′′, etc. have roughly the same size, and similarly for F, F′, etc. This technique (which is sometimes known as the “iterated popularity argument”) appears explicitly in work on the Radon transform [2], [9], and is also implicit in many works on the Kakeya problem, see e.g. [1], [3], [4], [6], [10], [11].

However, there is a slightly different approach to obtain both uniformity assumptions without refinement, which comes from trying to extremize (1) with respect to the sets E and F. The point is that if T χ_{E} is very small on some portion on F, then we can remove that portion from F, and the right-hand side of (1) will decrease faster than the left-hand side. The upshot of all this is that we can assume that

T χ_{E} ≳ α  (4)

on all of F and

T^{∗}χ_{F} ≳ β  (5)

on all of E, where α, β are given by (2), (3). We can also control the exceptional sets where T χ_{E} ≫ α and T^{∗}χ_{F} ≫ β by similar reasoning (if, for instance, the former set is too large, then we can restrict F to this exceptional set and get a better example for (1)).

In summary, the Extremization principle obtains results comparable to other techniques such as pigeonholing, but its advantage is one does not need to pass to subsets to enjoy the conclusions of the principle, they hold without exception because even a small exceptional set could be removed to contradict extremality.

There is only one major downside to using the Extremization principle - one loses the freedom of mobility of the x variable, or whatever variables are being optimized. For instance, in the previous application, once one tries to refine E or F the properties (4), (5) will be lost. In particular, it is difficult to combine the Extremization principle with the Dyadic Pigeonhole principle, and one must make a choice of one or the other. In practice the Dyadic Pigeonhole principle seems to be more useful for the majority of applications, though not for all.

Covering lemmas are usually proven using stopping time arguments, but they can often be done by extremization arguments as well, and sometimes one gets slightly better properties out of the covering in this fashion. Here’s an example in which one wishes to cover a set E efficiently by an “α-dimensional cover”.

Proposition 1.1. Let E be a finite union of dyadic unit cubes in R^{n}, and let 0 < α < n. Then we can find a finite collection Q of disjoint dyadic cubes Q of side-length l(Q) ≥ 1 covering E such that

• For all dyadic cubes I, we have

∑_{Q∈Q:Q⊂I} l(Q)^{α} ≤ l(I)^{α}  (6)

where l(Q) denotes the side-length of Q.

• We have

∑_{Q∈Q} l(Q)^{α} ∼ h_{α}(E),  (7)

where h_{α}(E) is the Hausdorff content of E at dimension α, i.e. the infimum of ∑_{i} r_{i}^{α} over all coverings of E by balls B(x_{i}, r_{i}).

• For all Q ∈ Q, we have

|E ∩ Q| ≥ l(Q)^{α}.  (8)

More generally, we have

|E ∩ Q| ≳ h_{α}(E ∩ Q).  (9)

Proof Define Q to be the collection of dyadic cubes of side-length l(Q) ≥ 1 which covers E and which minimizes the left-hand side of (7) subject to the above constraints. If (6) fails for some I, then we could replace all the cubes Q ∈ Q, Q ⊂ I with I in Q, contradicting minimality. Similarly, if (8) failed for some Q, then we could replace Q by all the unit cubes in E ∩Q, again contradicting minimality. The other properties are proven similarly and are left as an exercise to the reader.

Note that a stopping argument construction may give one of (8) or (6) (depending on whether one chooses an upward or downward stopping time), but not both simultaneously. In general, extremization techniques can do many of the things that more conventional recursive algorithms such as stopping time arguments do. These algorithms generally construct successive approximations x_{1}, x_{2}, … to a final desired object x, whereas the extremization technique jumps to x directly by extremizing an appropriate functional. (As mentioned before, the question of proving the existence of a minimum is usually quite easy to do (or get around) and is always rather “soft”, requiring very few hard estimates). The disadvantage of avoiding a direct construction and passing to the extremum is that one has no information on x other than that it extremizes the functional, so most of the effort goes into tailoring the functional in order that its extrema have all the desired properties. Whereas a recursive algorithm generally gives far more control on the limiting object x, with many opportunities to prove whatever properties are desired on x. Plus, it is often easier to tailor a recursive algorithm to one’s purposes than a single functional (at the cost of making convergence more difficult to prove).

For an application of this type of covering lemma, see [5]. Notice that the extremality hypothesis is easier to apply in the dyadic setting than it would in a continuous model. (Try proving the above proposition with dyadic cubes replaced by balls and allowing some loss of constant factors without using the dyadic model as an initial step). It seems that playing with extrema is more easily done in the precise dyadic setting than in the fuzzy continuous setting.

We now apply a similar technique to the Kakeya problem, obtaining “two ends” conditions which are the analogue of (8). One formulation of this problem is as follows. Suppose 0 < δ ≪ 1, and that T is a collection of δ × 1 tubes in R^{n} whose directions form a δ-separated set of directions. Let 0 < λ ≤ 1, and let E be a set in R^{n} such that

|E ∩ T | ≥ λδ^{n−1}  (10)

for all tubes T ∈ T. The Kakeya conjecture states that

|E| ≳ δ^{n−1−ε}λ^{n}(#T)  (11)

for all ε > 0 (with the implicit constants of course depending on ε). This conjecture is proven for n = 2, but only partial progress is known for n > 2.

Judging from (10), the set E ∩ T could behave in a number of ways, ranging from a λ × δ sub-tube of T, to an equally spaced collection of λ/δ δ-balls filling T out evenly, to a δ-discretized Cantor set of dimension 1 − log_{δ} λ. (Of course, these distinctions are only relevant when λ ≪ 1, since there is essentially only one set E ∩ T can be if λ ∼ 1). It is probably the latter type of set which is the most important in the Kakeya problem, which is one reason there is interest in Furstenburg β-sets (sets which contain a β-dimensional subset of a line in each direction). Unfortunately, we cannot reduce completely to this case as yet; however, we can make one step in this direction, namely the “two ends reduction” of Wolff [10], which at least eliminates the first type of set considered above.

Proposition 1.2. [10] To prove the Kakeya conjecture (11), it suffices to do so under the additional hypothesis that

|E ∩ T ∩ B(x, r)| ≲ λr^{ε/C}δ^{n−1}  (12)

for all balls B(x, r) of radius δ ≤ r ≪ 1.

Proof Let’s fix a single choice of r, and just obtain the reduction for this value of r. (The general case follows by iterating this, because there are very few distinct values of r we need to consider (log 1/δ if we are using dyadic scales, or we can whittle this down to C/ε if we are more astute and observe we only need consider r of the form δ^{kε/C})).

Fix ε > 0 and δ_{0} > 0; it of course suffices to show (11) for all δ > δ_{0}, since δ_{0} was arbitrary. Let A denote the best constant such that

|E| ≥ Aδ^{n−1−ε}λ^{n}(#T)  (13)

for all E, λ, δ, T obeying (10) and δ ≥ δ_{0}. Note that A is well-defined and positive because of our restriction δ ≥ δ_{0}. To prove (11) it suffices to prove it for an extreme value, so that

|E| = Aδ^{n−1−ε}λ^{n}(#T).  (14)

(Or we can get within epsilon of the optimal, that’s also okay for this argument). Our task is to show that A is bounded away from zero independently of δ_{0}.

Fix E, λ, ε, T. Call a tube T bad if (12) fails for some x. If fewer than half the tubes in T are bad then we are done, because we simply apply our hypothesis to the good tubes to obtain the desired lower bound for A. Now suppose that at least half the tubes in T are bad. This gives us at least #T/2 tubes T′ of dimensions δ ×Cr, each of which contains a subset of E of measure ≫ λr^{ε/C}δ^{n−1}. These tubes are only δ-separated in angle, but one can refine this to δr separated by reducing the number of tubes to about ∼ r^{n−1}#T. By applying a rescaled version of (13) we thus have

|E| ≫ Ar^{n}(δ/r)^{n−1−ε}(λr^{ε/C}/r)^{n}(r^{n−1}#T)

and this contradicts (14) if C is chosen appropriately.

Note the role of the ε in the above argument. Without the epsilon, (12) is trivial, and does not prevent E ∩T from concentrating on one end of T. Roughly speaking, the ε keeps (11) from being perfectly “scale invariant”, and makes the passing to a smaller scale just very slightly in our favour. Thus we can exclude one-ended sets E ∩ T because they can take advantage of a more favourable estimate. More generally, it is sometimes convenient to introduce a penalty term similar to the δ^{−ε} term in order to make the estimates for any situation other than the one we want to reduce to slightly more favourable than the default estimates. (Unlike most other techniques, the extremization technique can be very sensitive to very slight variations in the functional, especially if there are some sort of invariances or symmetries that one can break).

Another application of the extremization principle to Kakeya problems occurs in [8], in which it is a little inconvenient (though not impossible) to use the dyadic pigeonhole principle due to the bilinear formulation of problems in that paper. Another use will appear in [9].

References

[1] J. Bourgain, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. and Funct. Anal. 22 (1991), 147–187.

[2] M. Christ, Curvature, combinatorics, and convolution: a case study, preprint.

[3] N. Katz, I. Łaba, T. Tao, An improved bound for the Minkowski dimension of Besicovitch sets in R^{3}, submitted to Annals of Math.

[4] N. Katz, T. Tao, Bounds on arithmetic projections, and applications to the Kakeya conjecture, Math Res. Letters 6 (1999), 1-6.

[5] N. Katz, T. Tao, Some connections between the conjectures of Falconer, Furstenburg, and Erdös, in preparation.

[6] I. Łaba, T. Tao, An x-ray estimate in R^{n}, submitted, Revista Mat. Iber.

[7] T. Tao, Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates, submitted, Math Z.

[8] T. Tao, The dyadic pigeonhole principle, expository note.

[9] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), pp. 967–1000.

[10] T. Tao, J. Wright, L^{p} improving properties of averages over curves, submitted.

[11] T. Wolff: An improved bound for Kakeya type maximal functions, Revista Mat. Iberoamericana 11(1995), 651–674.

[12] T. Wolff: A mixed norm estimate for the x-ray transform, Revista Mat. Iberoamericana 14(1998), 561–601.

Department of Mathematics, UCLA, Los Angeles, CA 90024

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